mmtheorems70 - 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 10601-10700 108 10701-10781

Statement List for Metamath Proof Explorer - 6901-7000 - Page 70 of 108

Type	Label	Description
Statement
Theorem	abs3lem 6901	Lemma involving absolute value of differences.
|- A e. CC   &   |- B e. CC   &   |- C e. CC   &   |- D e. RR   =>   |- (((abs` (A - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (A - B)) < D)
Theorem	abs2dift 6902	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ((A e. CC /\ B e. CC) -> ((abs` A) - (abs` B)) <_ (abs` (A - B)))
Theorem	abs2difabst 6903	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
|- ((A e. CC /\ B e. CC) -> (abs` ((abs` A) - (abs` B))) <_ (abs` (A - B)))
Theorem	abs1m 6904	For any complex number, there exists a unit-magnitude multiplier that produces its absolute value. Part of proof of Theorem 13-2.12 of [Gleason] p. 195.
|- A e. CC   =>   |- E.x e. CC ((abs` x) = 1 /\ (abs` A) = (x x. A))
Theorem	recant 6905	Cancellation law involving the real part of a complex number.
|- ((A e. CC /\ B e. CC) -> (A.x e. CC (Re` (x x. A)) = (Re` (x x. B)) <-> A = B))
Theorem	absf 6906	Mapping domain and codomain of the absolute value function.
|- abs:CC-->RR
Theorem	abs3lemt 6907	Lemma involving absolute value of differences.
|- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. RR)) -> (((abs` (A - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (A - B)) < D))
Theorem	abslem2i 6908	Lemma involving absolute values.
|- A e. CC   &   |- A =/= 0   =>   |- (((*` (A / (abs` A))) x. A) + ((A / (abs` A)) x. (*` A))) = (2 x. (abs` A))
Theorem	abslem2 6909	Lemma involving absolute values.
|- A e. CC   =>   |- (A =/= 0 -> (((*` (A / (abs` A))) x. A) + ((A / (abs` A)) x. (*` A))) = (2 x. (abs` A)))
Theorem	seq1bnd 6910	An initial segment of an infinite sequence of complex numbers is bounded.
|- F:NN-->CC   =>   |- (A e. NN -> E.x e. RR A.y e. NN (y <_ A -> (abs` (F` y)) < x))
Theorem	seq1ublem 6911	Lemma for seq1ub 6912.
Theorem	seq1ub 6912	An upper bound for an initial segment of a sequence of reals.
|- F:NN-->RR   =>   |- ((A e. NN /\ B e. NN /\ A <_ B) -> (F` A) <_ sup(ran ( F |` {x e. NN | x <_ B}), RR, < ))
Theorem	cau2 6913	Two ways to express that a sequence meets the Cauchy criterion. Remark in [Gleason] p. 181. R can be either < or <_.
|- F:NN-->CC   &   |- (x e. RR -> (0 < x -> 0Rx))   =>   |- (A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y < z -> (abs` ((F` z) - (F` y)))Rx)) <-> A.x e. RR (0 < x -> E.y e. NN A.z e. NN (y <_ z -> (abs` ((F` z) - (F` y)))Rx)))
Theorem	cau3i 6914	A relationship used to derive two ways to express a Cauchy sequence.
|- Z (_ ZZ   =>   |- (E.m e. Z A.j e. Z A.k e. W ((m <_ j /\ m <_ k) -> ph) -> E.j e. Z A.k e. W (j <_ k -> ph))
Theorem	cau3ir 6915	A relationship used to derive two ways to express a Cauchy sequence. Normally Z and W are subsets of ZZ, and R is <_ or <. ph is ph(j, k, x).
|- (k = m -> (ph <-> ps))   &   |- (j = k -> (ps <-> ch))   &   |- (x = (y / 2) -> ((ph /\ ps) <-> th))   &   |- (x = y -> (ch <-> ta))   &   |- (((et /\ y e. RR) /\ (j e. Z /\ k e. W /\ m e. W)) -> (th -> ta))   =>   |- (et -> (A.x e. RR (0 < x -> E.j e. Z A.k e. W (jRk -> ph)) -> A.x e. RR (0 < x -> E.m e. Z A.j e. W A.k e. W ((mRj /\ mRk) -> ph))))
Theorem	cau3 6916	A relationship used to derive two ways to express a Cauchy sequence. ph is ph(j, k, x).
|- (k = m -> (ph <-> ps))   &   |- (j = k -> (ps <-> ch))   &   |- (x = (y / 2) -> ((ph /\ ps) <-> th))   &   |- (x = y -> (ch <-> ta))   &   |- (((et /\ y e. RR) /\ (j e. Z /\ k e. Z /\ m e. Z)) -> (th -> ta))   &   |- Z (_ ZZ   =>   |- (et -> (A.x e. RR (0 < x -> E.j e. Z A.k e. Z (j <_ k -> ph)) <-> A.x e. RR (0 < x -> E.m e. Z A.j e. Z A.k e. Z ((m <_ j /\ m <_ k) -> ph))))
Theorem	cau5i 6917	A relationship used to derive two ways to express a Cauchy sequence. ph is ph(j, k).
|- M e. ZZ   &   |- Z = (ZZ>` M)   &   |- W (_ ZZ   &   |- U (_ ZZ   =>   |- (E.m e. ZZ A.j e. ZZ A.k e. ZZ ((m <_ j /\ m <_ k) -> ph) -> E.m e. Z A.j e. W A.k e. U ((m <_ j /\ m <_ k) -> ph))
Theorem	cau4i 6918	A relationship used to derive two ways to express a Cauchy sequence. ph is ph(j, k).
|- M e. ZZ   &   |- Z = (ZZ>` M)   =>   |- (E.j e. Z A.k e. Z (j <_ k -> ph) -> E.j e. ZZ A.k e. ZZ (j <_ k -> ph))
Theorem	cau5 6919	A relationship used to derive two ways to express a Cauchy sequence. ph is ph(j, k).
|- M e. ZZ   &   |- Z = (ZZ>` M)   =>   |- (E.m e. ZZ A.j e. ZZ A.k e. ZZ ((m <_ j /\ m <_ k) -> ph) <-> E.m e. Z A.j e. Z A.k e. Z ((m <_ j /\ m <_ k) -> ph))
Theorem	cvg1i 6920	A relationship used to derive two ways to express that a sequence converges. Unlike cvg1 6921, j may be free in ph, so this can also be used for Cauchy sequences.
|- Z = (ZZ>` M)   =>   |- (E.j e. Z A.k e. Z (j <_ k -> ph) -> E.j e. Z A.k e. Z (j < k -> ph))
Theorem	cvg1 6921	A relationship used to derive two ways to express that a sequence converges. ph is ph(k).
|- Z = (ZZ>` M)   =>   |- (E.j e. Z A.k e. Z (j < k -> ph) <-> E.j e. Z A.k e. Z (j <_ k -> ph))
Theorem	cvg2 6922	Two ways to express that a sequence converges or is Cauchy.
|- ((y e. F /\ z e. G) -> A e. RR)   =>   |- (A.x e. RR (0 < x -> E.y e. F A.z e. G (ph -> A < x)) <-> A.x e. RR (0 < x -> E.y e. F A.z e. G (ph -> A <_ x)))
Theorem	cvg3 6923	A relationship used to derive two ways to express convergence. ph is ph(k).
|- M e. ZZ   &   |- (ZZ>` M) (_ Z   &   |- Z (_ ZZ   &   |- N e. ZZ   &   |- (ZZ>` N) (_ W   &   |- W (_ ZZ   =>   |- (E.j e. ZZ A.k e. ZZ (j <_ k -> ph) <-> E.j e. Z A.k e. W (j <_ k -> ph))
Theorem	cvganz 6924	Equivalence that lets us conjoin the properties of two independent converging sequences. k may be free in ph and ps. Compare r19.40 1762, where the implication holds in only one direction.
|- ((E.j e. ZZ A.k e. ZZ (j <_ k -> ph) /\ E.j e. ZZ A.k e. ZZ (j <_ k -> ps)) <-> E.j e. ZZ A.k e. ZZ (j <_ k -> (ph /\ ps)))
Theorem	cvganuz 6925	Lemma that lets us combine the properties of two independent converging sequences. k may be free in ph and ps.
|- M e. ZZ   &   |- Z = (ZZ>` M)   =>   |- ((E.j e. Z A.k e. Z (j <_ k -> ph) /\ E.j e. Z A.k e. Z (j <_ k -> ps)) <-> E.j e. Z A.k e. Z (j <_ k -> (ph /\ ps)))
Theorem	caubnd 6926	A Cauchy sequence of complex numbers is bounded.
|- F:NN-->CC   &   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((F` y) - (F` w))) < z))   =>   |- E.x e. RR A.y e. NN (abs` (F` y)) < x
Theorem	caure 6927	The real part of a complex Caucy sequence is a Cauchy sequence.
|- F:NN-->CC   &   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((F` y) - (F` w))) < z))   &   |- G Fn NN   &   |- (x e. NN ->