mmtheorems88 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 8701-8800 - Page 88 of 107

Type	Label	Description
Statement
Theorem	efper 8701	The exponential function is periodic. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ K e. ZZ) -> (exp` (A + ((i x. (2 x. pi)) x. K))) = (exp` A))
Theorem	eff1o 8702	The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (exp |` {x e. CC | (Im` x) e. (-upi[,)pi)}):{x e. CC | (Im` x) e. (-upi[,)pi)}-1-1-onto->(CC \ {0})
The natural logarithm on complex numbers
Syntax	clog 8703	Extend class notation with the natural logarithm function on complex numbers.
class log
Definition	df-log 8704	Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function").
|- log = `'(exp |` {x e. CC | (Im` x) e. (-upi[,)pi)})
Theorem	logrn 8705	The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class abstraction as simply ran log. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ran log = {x e. CC | (Im` x) e. (-upi[,)pi)}
Theorem	dflog2 8706	The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.)
|- log = `'(exp |` ran log)
Theorem	resslogrn 8707	The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- RR (_ ran log
Theorem	eff1o2 8708	The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (exp |` ran log):ran log-1-1-onto->(CC \ {0})
Theorem	logf1o 8709	The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008.)
|- log:(CC \ {0})-1-1-onto->ran log
Theorem	dfrelog 8710	The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (log |` RR+) = `'(exp |` RR)
Theorem	relogf1o 8711	The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (log |` RR+):RR+-1-1-onto->RR
Theorem	logclt 8712	Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ A =/= 0) -> (log` A) e. ran log)
Theorem	relogclt 8713	Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR+ -> (log` A) e. RR)
Theorem	eflogt 8714	Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ A =/= 0) -> (exp` (log` A)) = A)
Theorem	reeflogt 8715	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR+ -> (exp` (log` A)) = A)
Theorem	logeft 8716	Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- (A e. ran log -> (log` (exp` A)) = A)
Theorem	relogeft 8717	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (A e. RR -> (log` (exp` A)) = A)
Theorem	logeftb 8718	Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ A =/= 0 /\ B e. ran log) -> ((log` A) = B <-> (exp` B) = A))
Theorem	relogeftb 8719	Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR) -> ((log` A) = B <-> (exp` B) = A))
Theorem	log1 8720	The natural logarithm of 1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (log` 1) = 0
Theorem	loge 8721	The natural logarithm of e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (log` e) = 1
Theorem	pilog 8722	Relationship between pi and the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- pi = (i x. (log` -u1))
Theorem	relogoprlem 8723	Lemma for relogmult 8724 and relogdivt 8725. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2").
Theorem	relogmult 8724	The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (log` (A x. B)) = ((log` A) + (log` B)))
Theorem	relogdivt 8725	The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (log` (A / B)) = ((log` A) - (log` B)))
Theorem	explogt 8726	Exponentiation of a nonzero complex number to a nonnegative integer power. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((A e. CC /\ A =/= 0 /\ N e. NN0) -> (A^N) = (exp` (N x. (log` A))))
Theorem	reexplogt 8727	Exponentiation of a positive real number to a nonnegative integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ N e. NN0) -> (A^N) = (exp` (N x. (log` A))))
Theorem	relogexpt 8728	The natural logarithm of positive A raised to an nonnegative integer power. Property 4 of [Cohen] p. 301-302, restricted to natural logarithms and nonnegative-integer powers N. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ N e. NN0) -> (log` (A^N)) = (N x. (log` A)))
Theorem	relogiso 8729	The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- (log |` RR+) Isom < , < (RR+, RR)
Theorem	logltbt 8730	The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (A < B <-> (log` A) < (log` B)))
ZFC Set Theory plus Grothendieck's Axiom
Introduce Grothendieck's Axiom
Axiom	ax-groth 8731	Grothendieck's Axiom. For every set x there is an inaccessible cardinal y such that y is not in x. The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics." This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols - see grothprim 8737. An open problem is finding a shorter equivalent.
|- E.y(x e. y /\ A.z e. y (A.w(w (_ z -> w e. y) /\ E.w e. y A.v(v (_ z -> v e. w)) /\ A.z(z (_ y -> (z ~~ y \/ z e. y)))
Theorem	axgroth2 8732	Alternate version of Grothendieck's Axiom.
|- E.y(x e. y /\ A.z e. y (A.w(w (_ z -> w e. y) /\ E.w e. y A.v(v (_ z -> v e. w)) /\ A.z(z (_ y -> (y ~<_ z \/ z e. y)))
Theorem	axgroth3 8733	Alternate version of Grothendieck's Axiom. ax-ac 4727 is used to derive this version.
|- E.y(x e. y /\ A.z e. y (A.w(w (_ z -> w e. y) /\ E.w e. y A.v(v (_ z -> v e. w)) /\ A.z(z (_ y -> ((y \ z) ~<_ z \/ z e. y)))
Theorem	axgroth4 8734	Alternate version of Grothendieck's Axiom. ax-ac 4727 is used to derive this version.
|- E.y(x e. y /\ A.z e. y E.v e. y A.w(w (_ z -> w e. (y i^i v)) /\ A.z(z (_ y -> ((y \ z) ~<_ z \/ z e. y)))
Theorem	grothinf 8735	Grothendieck's Axiom implies the Axiom of Infinity (in the form of omex 4610). Note that our proof does not depend on the Axiom of Infinity.
|- om e. V
Theorem	grothprimlem 8736	Lemma for grothprim 8737. Expand the membership of an unordered pair into primitives.
Theorem	grothprim 8737	Grothendieck's Axiom ax-groth 8731 expanded into set theory primitives using 163 symbols. An open problem is whether a shorter equivalent exists (when expanded to primitives).
|- E.y(x e. y /\ A.z((z e. y -> E.v(v e. y /\ A.w(A.u(u e. w -> u e. z) -> (w e. y /\ w e. v)))) /\ E.w((w e. z -> w e. y) -> (A.v((v e. z -> E.tA.u(E.g(g e. w /\ A.h(h e. g <-> (h = v \/ h = u))) -> u = t)) /\ (v e. y -> (v e. z \/ E.u(u e. z /\ E.g(g e. w /\ A.h(h e. g <-> (h = u \/ h = v))))))) \/ z e. y))))
Humor
April Fool's theorem
Theorem	avril1 8738	Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid germanus dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object x equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) A reply to skeptics can be found at http://us.metamath.org/mpegif/mmnotes.txt, under the 1-Apr-2006 entry.
|- -. (AP~RR(i` 1) /\ F(/)(0 x. 1))
Theorem	2bornot2b 8739	The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.)
|- (2 x. B \/ -. 2 x. B)
Theorem	helloworld 8740	The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.)
|- -. (h e. (LL0) /\ W(/)(R.1d))
Theorem	1p1e2 8741	One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.)
|- (1 + 1) = 2
Hilbert Space Explorer
Syntax	chil 8742	Extend class notation with Hilbert vector space.
class H~
Syntax	cva 8743	Extend class notation with vector addition in Hilbert space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 5220.
class +h
Syntax	csm 8744	Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
class .h
Syntax	c0v 8745	Extend class notation with zero vector in Hilbert space.
class 0h
Syntax	cmv 8746	Extend class notation with vector subtraction in Hilbert space.
class -h
Syntax	csp 8747	Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of A and B is usually written <.A, B>. but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 2413.
class .ih
Syntax	cno 8748	Extend class notation with the norm function in Hilbert space. In the literature, the norm of A is usually written "|| A ||", but we use function notation to take advantage of our existing theorems about functions.
class normh
Syntax	ccau 8749	Extend class notation with set of Cauchy sequences in Hilbert space.
class Cauchy
Syntax	chli 8750	Extend class notation with convergence relation in Hilbert space.
class ~~>v
Syntax	csh 8751	Extend class notation with set of subspaces of a Hilbert space.
class SH
Syntax	cch 8752	Extend class notation with set of closed subspaces of a Hilbert space.
class CH
Syntax	cort 8753	Extend class notation with orthogonal complement in CH.
class _|_
Syntax	cph 8754	Extend class notation with subspace sum in CH.
class +H
Syntax	cspn 8755	Extend class notation with subspace span in CH.
class span
Syntax	chj 8756	Extend class notation with join in CH.
class vH
Syntax	chsup 8757	Extend class notation with supremum of a collection in CH.
class \/H
Syntax	c0h 8758	Extend class notation with zero of CH.
class 0H
Syntax	ccm 8759	Extend class notation with the commutes relation on a Hilbert lattice.
class C_H
Syntax	cpj 8760	Extend class notation with set of projections on a Hilbert space.
class proj
Syntax	chos 8761	Extend class notation with sum of Hilbert space operators.
class +op
Syntax	chot 8762	Extend class notation with scalar product of a Hilbert space operator.
class .op
Syntax	chod 8763	Extend class notation with difference of Hilbert space operators.
class -op
Syntax	chfs 8764	Extend class notation with sum of Hilbert space functionals.
class +fn
Syntax	chft 8765	Extend class notation with scalar product of Hilbert space functional.
class .fn
Syntax	ch0o 8766	Extend class notation with the Hilbert space zero operator.
class 0hop
Syntax	chio 8767	Extend class notation with Hilbert space identity operator.
class Iop
Syntax	cnop 8768	Extend class notation with the operator norm function.
class normop
Syntax	cco 8769	Extend class notation with set of continuous Hilbert space operators.
class ConOp
Syntax	clo 8770	Extend class notation with set of linear Hilbert space operators.
class LinOp
Syntax	cbo 8771	Extend class notation with set of bounded linear operators.
class BndLinOp
Syntax	cuo 8772	Extend class notation with set of unitary Hilbert space operators.
class UniOp
Syntax	cho 8773	Extend class notation with set of Hermitian Hilbert space operators.
class HrmOp
Syntax	cnmf 8774	Extend class notation with the functional norm function.
class normfn
Syntax	cnl 8775	Extend class notation with the functional nullspace function.
class null
Syntax	ccnf 8776	Extend class notation with set of continuous Hilbert space functionals.
class ConFn
Syntax	clf 8777	Extend class notation with set of linear Hilbert space functionals.
class LinFn
Syntax	cado 8778	Extend class notation with Hilbert space adjoint function.
class adjh
Syntax	cbr 8779	Extend class notation with the bra of a vector in Dirac bra-ket notation.
class bra
Syntax	ck 8780	Extend class notation with the outer product of two vectors in Dirac bra-ket notation.
class ketbra
Syntax	cleo 8781	Extend class notation with positive operator ordering.
class <_op
Syntax	cei 8782	Extend class notation with Hilbert space eigenvector function.
class eigvec
Syntax	cel 8783	Extend class notation with Hilbert space eigenvalue function.
class eigval
Syntax	cspc 8784	Extend class notation with the spectrum of an operator.
class Lambda
Syntax	cst 8785	Extend class notation with set of states on a Hilbert lattice.
class States
Syntax	chst 8786	Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.
class CHStates
Syntax	cat 8787	Extend class notation with set of atoms on a Hilbert lattice.
class Atoms
Syntax	ccv 8788	Extend class notation with the covers relation on a Hilbert lattice.
class <o
Syntax	cmd 8789	Extend class notation with