mmtheorems70 - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	absf 6901	Mapping domain and codomain of the absolute value function.
|- abs:CC-->RR
Theorem	abs3lemt 6902	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 6903	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 6904	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 6905	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 6906	Lemma for seq1ub 6907.
Theorem	seq1ub 6907	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 6908	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 6909	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 6910	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 6911	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 6912	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 6913	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 6914	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 6915	A relationship used to derive two ways to express that a sequence converges. Unlike cvg1 6916, 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 6916	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 6917	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 6918	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 6919	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 6920	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 6921	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 6922	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 -> (G` x) = (Re` (F` x)))   =>   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((G` y) - (G` w))) < z))
Theorem	cauim 6923	The imaginary 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 -> (G` x) = (Im` (F` x)))   =>   |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((G` y) - (G` w))) < z))
Theorem	ser1absdiflem 6924	Lemma for ser1absdif 6925. Warning: The HTML proof page is 1/2 megabyte in size.
Theorem	ser1absdif 6925	Generalized triangle inequality: absolute value of a partial sum is less than or equal to sum of absolute values.
|- F:NN-->CC   &   |- G Fn NN   &   |- (x e. NN -> (G` x) = (abs` (F` x)))   =>   |- ((A e. NN /\ B e. NN /\ A < B