mmtheorems50 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10687

Statement List for Metamath Proof Explorer - 4901-5000 - Page 50 of 107

Type	Label	Description
Statement
Theorem	cf0 4901	Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102.
|- (cf` (/)) = (/)
Theorem	cardcf 4902	Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.
|- (card` (cf` A)) = (cf` A)
Theorem	cflecard 4903	Cofinality is bounded by the cardinality of its argument.
|- (cf` A) (_ (card` A)
Theorem	cfle 4904	Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102.
|- (cf` A) (_ A
Theorem	cfeq0 4905	Only the ordinal zero has cofinality zero.
|- (A e. On -> ((cf` A) = (/) <-> A = (/)))
Theorem	cfsuc 4906	Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102.
|- (A e. On -> (cf` suc A) = 1o)
Theorem	cfom 4907	Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102.
|- (cf` om) = om
Cardinal number arithmetic
Syntax	ccda 4908	Extend class definition to include cardinal number addition.
class +c
Definition	df-cda 4909	Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 4911 for its value and a description.
|- +c = {<.<.x, y>., z>. | z = ((x X. {(/)}) u. (y X. {1o}))}
Theorem	cdavalt 4910	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258.
|- ((A e. C /\ B e. D) -> (A +c B) = ((A X. {(/)}) u. (B X. {1o})))
Theorem	cdaval 4911	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while cross product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 4822, carddom 4827, and cardsdom 4828. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available.
|- A e. V   &   |- B e. V   =>   |- (A +c B) = ((A X. {(/)}) u. (B X. {1o}))
Theorem	uncdadom 4912	Cardinal addition dominates union.
|- A e. V   &   |- B e. V   =>   |- (A u. B) ~<_ (A +c B)
Theorem	cdaun 4913	Cardinal addition is equinumerous to union for disjoint sets.
|- A e. V   &   |- B e. V   =>   |- ((A i^i B) = (/) -> (A +c B) ~~ (A u. B))
Theorem	pm110.643 4914	1+1=2 for cardinal number addition. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Unlike us, Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 4717), but after applying definitions, our theorem is equivalent. See also the comment for pm54.43 4563. The comment for cdaval 4911 explains why we use ~~ instead of =.
|- (1o +c 1o) ~~ 2o
Theorem	cdaen 4915	Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- ((A ~~ B /\ C ~~ D) -> (A +c C) ~~ (B +c D))
Theorem	cda0en 4916	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143.
|- A e. V   =>   |- (A +c (/)) ~~ A
Theorem	cda1en 4917	Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143.
|- A e. V   =>   |- (A +c 1o) ~~ suc (card` A)
Theorem	xp1en 4918	One times a cardinal number.
|- A e. V   =>   |- (A X. 1o) ~~ A
Theorem	xp2cda 4919	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258.
|- A e. V   =>   |- (A X. 2o) = (A +c A)
Theorem	cdacomen 4920	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   =>   |- (A +c B) ~~ (B +c A)
Theorem	cdaassen 4921	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A +c B) +c C) ~~ (A +c (B +c C))
Theorem	xpcdaen 4922	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A X. (B +c C)) ~~ ((A X. B) +c (A X. C))
Theorem	mapcdaen 4923	Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ^m (B +c C)) ~~ ((A ^m B) X. (A ^m C))
Theorem	cdadom1 4924	Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ~<_ B -> (A +c C) ~<_ (B +c C))
Theorem	cdadom2 4925	Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ~<_ B -> (C +c A) ~<_ (C +c B))
Theorem	cdadom3 4926	A set is dominated by its cardinal sum with another.
|- A e. V   &   |- B e. V   =>   |- A ~<_ (A +c B)
Theorem	cdafi 4927	The cardinal sum of two finite sets is finite.
|- ((A ~< om /\ B ~< om) -> (A +c B) ~< om)
Theorem	cdainf 4928	A set is infinite iff the cardinal sum with itself is infinite.
|- A e. V   =>   |- (om ~<_ A <-> om ~<_ (A +c A))
ZFC Axioms with no distinct variable requirements
Theorem	nd1 4929	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.x y e. z)
Theorem	nd2 4930	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.x z e. y)
Theorem	nd3 4931	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.z x e. y)
Theorem	nd4 4932	A lemma for proving conditionless ZFC axioms.
|- (A.x x = y -> -. A.z y e. x)
Theorem	nd5 4933	A lemma for proving conditionless ZFC axioms.
|- (-. A.y y = x -> (z = y -> A.x z = y))
Theorem	axextnd 4934	A version of the Axiom of Extensionality with no distinct variable conditions.
|- E.x((x e. y <-> x e. z) -> y = z)
Theorem	axrepndlem1 4935	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepndlem2 4936	Lemma for the Axiom of Replacement with no distinct variable conditions.
Theorem	axrepnd 4937	A version of the Axiom of Replacement with no distinct variable conditions.
|- E.x(E.yA.z(ph -> z = y) -> A.z(A.y z e. x <-> E.x(A.z x e. y /\ A.yph)))
Theorem	axunndlem1 4938	Lemma for the Axiom of Union with no distinct variable conditions.
Theorem	axunnd 4939	A version of the Axiom of Union with no distinct variable conditions.
|- E.xA.y(E.x(y e. x /\ x e. z) -> y e. x)
Theorem	axpowndlem1 4940	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem2 4941	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem3 4942	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpowndlem4 4943	Lemma for the Axiom of Power Sets with no distinct variable conditions.
Theorem	axpownd 4944	A version of the Axiom of Power Sets with no distinct variable conditions.
|- (-. x = y -> E.xA.y(A.x(E.z x e. y -> A.y x e. z) -> y e. x))
Theorem	axregndlem1 4945	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregndlem2 4946	Lemma for the Axiom of Regularity with no distinct variable conditions.
Theorem	axregnd 4947	A version of the Axiom of Regularity with no distinct variable conditions.
|- (x e. y -> E.x(x e. y /\ A.z(z e. x -> -. z e. y)))
Theorem	axinfndlem1 4948	Lemma for the Axiom of Infinity with no distinct variable conditions.
Theorem	axinfnd 4949	A version of the Axiom of Infinity with no distinct variable conditions.
|- E.x(y e. z -> (y e. x /\ A.y(y e. x -> E.z(y e. z /\ z e. x))))
Theorem	axacndlem1 4950	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem2 4951	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem3 4952	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem4 4953	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacndlem5 4954	Lemma for the Axiom of Choice with no distinct variable conditions.
Theorem	axacnd 4955	A version of the Axiom of Choice with no distinct variable conditions.
|- E.xA.yA.z(A.x(y e. z /\ z e. w) -> E.wA.y(E.w((y e. z /\ z e. w) /\ (y e. w /\ w e. x)) <-> y = w))
Theorem	zfcndext 4956	Axiom of Extensionality, reproved from conditionless ZFC version and predicate calculus.
|- (A.z(z e. x <-> z e. y) -> x = y)
Theorem	zfcndrep 4957	Axiom of Replacement, reproved from conditionless ZFC axioms.
|- (A.wE.yA.z(A.yph -> z = y) -> E.yA.z(z e. y <-> E.w(w e. x /\ A.yph)))
Theorem	zfcndun 4958	Axiom of Union, reproved from conditionless ZFC axioms.
|- E.yA.z(E.w(z e. w /\ w e. x) -> z e. y)
Theorem	zfcndpow 4959	Axiom of Power Sets, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 2768.
|- E.yA.z(A.w(w e. z -> w e. x) -> z e. y)
Theorem	zfcndreg 4960	Axiom of Regularity, reproved from conditionless ZFC axioms..
|- (E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x)))
Theorem	zfcndinf 4961	Axiom of Infinity, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets, we are justified in referencing theorem el 2747 in the proof.
|- E.y(x e. y /\ A.z(z e. y -> E.w(z e. w /\ w e. y)))
Theorem	zfcndac 4962	Axiom of Choice, reproved from conditionless ZFC axioms.
|- E.yA.zA.w((z e. w /\ w e. x) -> E.vA.u(E.t((u e. w /\ w e. t) /\ (u e. t /\ t e. y)) <-> u = v))
Real and complex numbers
Dedekind-cut construction of real and complex numbers
Syntax	cnpi 4963	The set of positive integers, which is the set of natural numbers om with 0 removed. Note: This is the start of the Dedekind-cut construction of real and complex numbers. The last lemma of the construction is mulcnsrec 5255. The actual set of Dedekind cuts is defined by df-np 5077.
class N.
Syntax	cpli 4964	Positive integer addition.
class +N
Syntax	cmi 4965	Positive integer multiplication.
class .N
Syntax	clti 4966	Positive integer ordering relation.
class <N
Syntax	cplpq 4967	Positive fraction pre-addition.
class +pQ
Syntax	cmpq 4968	Positive fraction pre-multiplication.
class .pQ
Syntax	ceq 4969	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 4970	Set of positive fractions.
class Q.
Syntax	c1q 4971	The positive fraction constant 1.
class 1Q
Syntax	cplq 4972	Positive fraction addition.
class +Q
Syntax	cmq 4973	Positive fraction multiplication.
class .Q
Syntax	crq 4974	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 4975	Positive fraction ordering relation.
class <Q
Syntax	cnp 4976	Set of positive reals.
class P.
Syntax	c1p 4977	Positive real constant 1.
class 1P
Syntax	cpp 4978	Positive real addition.
class +P.
Syntax	cmp 4979	Positive real multiplication.
class .P.
Syntax	cltp 4980	Positive real ordering relation.
class <P
Syntax	cplpr 4981	Signed real pre-addition.
class +pR
Syntax	cmpr 4982	Signed real pre-multiplication.
class .pR
Syntax	cer 4983	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 4984	Set of signed reals.
class R.
Syntax	c0r 4985	The signed real constant 0.
class 0R
Syntax	c1r 4986	The signed real constant 1.
class 1R
Syntax	cm1r 4987	The signed real constant -1.
class -1R
Syntax	cplr 4988	Signed real addition.
class +R
Syntax	cmr 4989	Signed real multiplication.
class .R
Syntax	cltr 4990	Signed real ordering relation.
class <R
Definition	df-ni 4991	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5231, and is intended to be used only by the construction.
|- N. = (om \ {(/)})
Definition	df-pli 4992	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5231, and is intended to be used only by the construction.
|- +N = ( +o |` (N. X. N.))
Definition	df-mi 4993	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5231, and is intended to be used only by the construction.
|- .N = ( .o |` (N. X. N.))
Definition	df-lti 4994	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 5231, and is intended to be used only by the construction.
|- <N = (E i^i (N. X. N.))
Theorem	elni 4995	Membership in the class of positive integers.
|- (A e. N. <-> (A e. om /\ A =/= (/)))