mmtheorems63 - 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 - 6201-6300 - Page 63 of 107

Type	Label	Description
Statement
Theorem	flval3t 6201	An alternate way to define the floor (greatest integer) function, as the supremum of all integers less than or equal to its argument.
|- (A e. RR -> (|_` A) = sup({x e. ZZ | x <_ A}, RR, < ))
Theorem	flwordit 6202	Ordering relationship for the greatest integer function.
|- ((A e. RR /\ B e. RR /\ A <_ B) -> (|_` A) <_ (|_` B))
Theorem	flbit 6203	A condition equivalent to floor.
|- ((A e. RR /\ B e. ZZ) -> ((|_` A) = B <-> (B <_ A /\ A < (B + 1))))
Theorem	flge0nn0t 6204	The floor of a number greater than or equal to 0 is a nonnegative integer.
|- ((A e. RR /\ 0 <_ A) -> (|_` A) e. NN0)
Theorem	flge1nnt 6205	The floor of a number greater than or equal to 1 is a natural number.
|- ((A e. RR /\ 1 <_ A) -> (|_` A) e. NN)
Theorem	fladdzt 6206	An integer can be moved in and out of the floor of a sum.
|- ((A e. RR /\ B e. ZZ) -> (|_` (A + B)) = ((|_` A) + B))
Theorem	btwnzge0t 6207	A real bounded between an integer and its successor is nonnegative iff the integer is nonnegative. Second half of Lemma 13-4.1 of [Gleason] p. 217. (For the first half see rebtwnz 6189.)
|- (((A e. RR /\ B e. ZZ) /\ (B <_ A /\ A < (B + 1))) -> (0 <_ A <-> 0 <_ B))
Theorem	flhalft 6208	Ordering relation for the floor of half of an integer.
|- (N e. ZZ -> N <_ (2 x. (|_` ((N + 1) / 2))))
Theorem	ceiclt 6209	The ceiling function returns an integer (closure law). (Contributed by Jeffrey Hankins, 10-Jun-2007.)
|- (A e. RR -> -u(|_` -uA) e. ZZ)
Theorem	ceiget 6210	The ceiling of a real number is greater than or equal to that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
|- (A e. RR -> A <_ -u(|_` -uA))
Theorem	ceim1lt 6211	One less than the ceiling of a real number is strictly less than that number. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
|- (A e. RR -> (-u(|_` -uA) - 1) < A)
Theorem	ceilet 6212	The ceiling of a real number is the smallest integer greater than or equal to it. (Contributed by Jeffrey Hankins, 10-Jun-2007.)
|- ((A e. RR /\ B e. ZZ /\ A <_ B) -> -u(|_` -uA) <_ B)
Rational numbers (as a subset of complex numbers)
Definition	df-q 6213	Define the set of rational numbers. Definition of rationals in [Apostol] p. 22.
|- QQ = {x | E.m e. ZZ E.n e. NN x = (m / n)}
Theorem	elq 6214	Membership in the set of rationals.
|- (A e. QQ <-> E.x e. ZZ E.y e. NN A = (x / y))
Theorem	znq 6215	The ratio of an integer and a natural number is a rational number.
|- ((A e. ZZ /\ B e. NN) -> (A / B) e. QQ)
Theorem	qret 6216	A rational number is a real number.
|- (A e. QQ -> A e. RR)
Theorem	zqt 6217	An integer is a rational number.
|- (A e. ZZ -> A e. QQ)
Theorem	zssq 6218	The integers are a subset of the rationals.
|- ZZ (_ QQ
Theorem	nn0ssq 6219	The nonnegative integers are a subset of the rationals.
|- NN0 (_ QQ
Theorem	nnssq 6220	The natural numbers are a subset of the rationals.
|- NN (_ QQ
Theorem	qssre 6221	The rationals are a subset of the reals.
|- QQ (_ RR
Theorem	qsscn 6222	The rationals are a subset of the complex numbers.
|- QQ (_ CC
Theorem	nnqt 6223	A natural number is rational.
|- (A e. NN -> A e. QQ)
Theorem	qcnt 6224	A rational number is a complex number.
|- (A e. QQ -> A e. CC)
Theorem	qex 6225	The set of rational numbers exists.
|- QQ e. V
Theorem	qaddclt 6226	Closure of addition of rationals.
|- ((A e. QQ /\ B e. QQ) -> (A + B) e. QQ)
Theorem	qnegclt 6227	Closure law for the negative of a rational.
|- (A e. QQ -> -uA e. QQ)
Theorem	qmulclt 6228	Closure of multiplication of rationals.
|- ((A e. QQ /\ B e. QQ) -> (A x. B) e. QQ)
Theorem	qsubclt 6229	Closure of subtraction of rationals.
|- ((A e. QQ /\ B e. QQ) -> (A - B) e. QQ)
Theorem	qrecclt 6230	Closure of reciprocal of rationals.
|- ((A e. QQ /\ A =/= 0) -> (1 / A) e. QQ)
Theorem	qdivclt 6231	Closure of division of rationals.
|- (((A e. QQ /\ B e. QQ) /\ B =/= 0) -> (A / B) e. QQ)
Theorem	qrevaddclt 6232	Reverse closure law for addition of rationals.
|- (B e. QQ -> ((A e. CC /\ (A + B) e. QQ) <-> A e. QQ))
Theorem	nnrecqt 6233	The reciprocal of a natural number is rational.
|- (A e. NN -> (1 / A) e. QQ)
Theorem	nnrecret 6234	The reciprocal of a natural number is real.
|- (A e. NN -> (1 / A) e. RR)
Theorem	qbtwnre 6235	The rational numbers are dense in RR: any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28.
|- ((A e. RR /\ B e. RR /\ A < B) -> E.x e. QQ (A < x /\ x < B))
Theorem	qbtwnxr 6236	The rational numbers are dense in RR*: any two extended real numbers have a rational between them.
|- ((A e. RR* /\ B e. RR* /\ A < B) -> E.x e. QQ (A < x /\ x < B))
Theorem	qsqueeze 6237	If a nonnegative real is less than any positive rational, it is zero.
|- ((A e. RR /\ 0 <_ A /\ A.x e. QQ (0 < x -> A < x)) -> A = 0)
Positive reals (as a subset of complex numbers)
Definition	df-rp 6238	Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20.
|- RR+ = {x e. RR | 0 < x}
Theorem	elrp 6239	Membership in the set of positive reals.
|- (A e. RR+ <-> (A e. RR /\ 0 < A))
Theorem	elrpi 6240	Membership in the set of positive reals.
|- A e. RR   &   |- 0 < A   =>   |- A e. RR+
Theorem	rpret 6241	A positive real is a real.
|- (A e. RR+ -> A e. RR)
Theorem	rpssre 6242	The positive reals are a subset of the reals.
|- RR+ (_ RR
Theorem	rpgt0t 6243	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
|- (A e. RR+ -> 0 < A)
Theorem	rpge0t 6244	A positive real is greater than or equal to zero.
|- (A e. RR+ -> 0 <_ A)
Theorem	ralrp 6245	Quantification over positive reals.
|- (A.x e. RR+ ph <-> A.x e. RR (0 < x -> ph))
Theorem	rpaddclt 6246	Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20.
|- ((A e. RR+ /\ B e. RR+) -> (A + B) e. RR+)
Theorem	rpmulclt 6247	Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20.
|- ((A e. RR+ /\ B e. RR+) -> (A x. B) e. RR+)
Theorem	rpdivclt 6248	Closure law for multiplication of positive reals. (Contributed by FL, 27-Dec-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (A / B) e. RR+)
Theorem	0nrp 6249	Zero is not a positive real. Axiom 9 of [Apostol] p. 20.
|- -. 0 e. RR+
Monotonic sequences
Theorem	monoord 6250	Ordering relation for a monotonic sequence.
|- F:NN-->RR   &   |- (x e. NN -> (F` x) <_ (F` (x + 1)))   =>   |- ((A e. NN /\ B e. NN /\ A <_ B) -> (F` A) <_ (F` B))
The infinite sequence builder "seq1"
Theorem	om2uz0 6251	The mapping G is a one-to-one mapping from om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number C (normally 0 for the upper integers NN0 or 1 for the upper integers NN), 1 maps to C + 1, etc. This theorem shows the value of G at ordinal natural number zero. (This series of theorems generalizes an earlier series for NN0 contributed by Raph Levien, 10-Apr-2004.)
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (G` (/)) = C
Theorem	om2uzsuc 6252	The value of G (see om2uz0 6251) at a successor.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (A e. om -> (G` suc A) = ((G` A) + 1))
Theorem	om2uzuz 6253	The value G (see om2uz0 6251) at an ordinal natural number is in the upper integers.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (A e. om -> (G` A) e. {z e. ZZ | C <_ z})
Theorem	om2uzlt 6254	Less-than relation for G (see om2uz0 6251).
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- ((A e. om /\ B e. om) -> (A e. B -> (G` A) < (G` B)))
Theorem	om2uzlt2 6255	The mapping G (see om2uz0 6251) preserves order.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- ((A e. om /\ B e. om) -> (A e. B <-> (G` A) < (G` B)))
Theorem	om2uzran 6256	Range of G (see om2uz0 6251).
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- ran G = {z e. ZZ | C <_ z}
Theorem	om2uzf1o 6257	G (see om2uz0 6251) is a one-to-one onto mapping.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- G:om-1-1-onto->{z e. ZZ | C <_ z}
Theorem	uzrdgval 6258	A helper lemma for the value of a recursive definition generator on upper integers (typically either NN or NN0) with characteristic function F and initial value A. Normally F is a function on the partition, and A is a member of the partition. See also comment in om2uz0 6251.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (B e. {z e. ZZ | C <_ z} -> ((rec(F, A) o. `'G)` B) = (rec(F, A)` (`'G` B)))
Theorem	uzrdgini 6259	Initial value of a recursive definition generator on upper integers. See comment in uzrdgval 6258.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (A e. B -> ((rec(F, A) o. `'G)` C) = A)
Theorem	uzrdgsuc 6260	Successor value of a recursive definition generator on upper integers. See comment in uzrdgval 6258.
|- C e. ZZ   &   |- G = (rec({<.x, y>. | y = (x + 1)}, C) |` om)   =>   |- (B e. {z e. ZZ | C <_ z} -> ((rec(F, A) o. `'G)` (B + 1)) = (F` ((rec(F, A) o. `'G)` B)