mmtheorems93 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 9201-9300 - Page 93 of 108

Type	Label	Description
Statement
Theorem	projlem16 9201	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Existence of a vector sequence member, used to show (via Axiom of Choice) the vector sequence existence. Used by projlem17 9202.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- C e. NN   =>   |- E.z e. H ((R - (1 / C)) < (normh` (z -h A)) /\ (normh` (z -h A)) < (R + (1 / C)))
Theorem	projlem17 9202	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). This uses the Axiom of Choice to show the existence of a vector sequence satisfying the assumption of Lemma 3.6 of [Beran] p. 100: "Let {yn } be a sequence of W such that i0 - 1/n < ||x0 - yn || < i0 + 1/n." Here, H corresponds to "W"; f:NN-->H to "{yn }"; w to "n"; R to "i0 "; and (norm` (A -h (f` w))) to "||x0 - yn ||". Used by projlem 9217.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   =>   |- E.f(f:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((f` w) -h A)) /\ (normh` ((f` w) -h A)) < (R + (1 / w))))
Theorem	projlem18 9203	Part of Lemma 3.6 of [Beran] p. 101, top. Used by projlem19 9204.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- B e. H   &   |- C e. H   =>   |- (4 x. (R^2)) <_ ((normh` ((B +h C) -h (2 .h A)))^2)
Theorem	projlem19 9204	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem20 9205.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- B e. H   &   |- C e. H   &   |- D e. NN   &   |- G e. NN   &   |- N e. NN   =>   |- (((normh` (B -h A)) < (R + (1 / D)) /\ (normh` (C -h A)) < (R + (1 / G))) -> ((N <_ D /\ N <_ G) -> (normh` (B -h C)) < (sqr` ((4 x. ((2 x. R) + 1)) / N))))
Theorem	projlem20 9205	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem27 9212.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- N e. NN   =>   |- (((B e. H /\ C e. H) /\ (D e. NN /\ G e. NN)) -> (((normh` (B -h A)) < (R + (1 / D)) /\ (normh` (C -h A)) < (R + (1 / G))) -> ((N <_ D /\ N <_ G) -> (normh` (B -h C)) < (sqr` ((4 x. ((2 x. R) + 1)) / N)))))
Theorem	projlem21 9206	Part of Lemma 3.6 of [Beran] p. 100. The hypothesis lets us work with our postulated vector sequence (whose existence was shown by projlem17 9202). Here we just show the sequence value belongs to the closed subspace H. Used by projlem27 9212 projlem28 9213.
|- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))   =>   |- (ph -> (D e. NN -> (F` D) e. H))
Theorem	projlem22 9207	Part of Lemma 3.6 of [Beran] p. 100. Here we show a member of the vector sequence is bounded. Used by projlem27 9212.
|- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))   =>   |- (ph -> (D e. NN -> (normh` ((F` D) -h A)) < (R + (1 / D))))
Theorem	projlem23 9208	Part of Lemma 3.6 of [Beran] p. 101. The hypothesis lets us work with the sequence G which corresponds to Beran's "{||yn-x0||}". Used by projlem25 9210 projlem26 9211.
|- G = {<.x, y>. | (x e. NN /\ y = (normh` ((F` x) -h A)))}   =>   |- (D e. NN -> (G` D) = (normh` ((F` D) -h A)))
Theorem	projlem24 9209	Part of Lemma 3.6 of [Beran] p. 101. Here we show our vector sequence implies the real numbers sequence G corresponding to Beran's "{||yn-x0||}". Used by projlem25 9210 projlem26 9211.
|- G = {<.x, y>. | (x e. NN /\ y = (normh` ((F` x) -h A)))}   &   |- A e. H~   =>   |- (F:NN-->H~ -> G:NN-->CC)
Theorem	projlem25 9210	Part of Lemma 3.6 of [Beran] p. 101. "The sequence {||yn-x0||} of real numbers converges to the number ||y0-x0||" (here, "y0" is A and "x0" is z). Used by projlem31 9216.
|- G = {<.x, y>. | (x e. NN /\ y = (normh` ((F` x) -h A)))}   &   |- A e. H~   &   |- F e. V   =>   |- (F ~~>v z -> G ~~> (normh` (z -h A)))
Theorem	projlem26 9211	Part of Lemma 3.6 of [Beran] p. 101. The real sequence G (Beran's "{||yn-x0||}") converges to the infimum of norms. Used by projlem31 9216.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))   &   |- G = {<.x, y>. | (x e. NN /\ y = (normh` ((F` x) -h A)))}   =>   |- (ph -> G ~~> R)
Theorem	projlem27 9212	Part of Lemma 3.6 of [Beran] p. 101. Boundedness to show F is a Cauchy sequence. Used by projlem28 9213.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))   &   |- N e. NN   =>   |- ((ph /\ (D e. NN /\ G e. NN)) -> ((N <_ D /\ N <_ G) -> (normh` ((F` D) -h (F` G))) < (sqr` ((4 x. ((2 x. R) + 1)) / N))))
Theorem	projlem28 9213	Part of Lemma 3.6 of [Beran] p. 101. Boundedness to show F is a Cauchy sequence. Used by projlem29 9214.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))   &   |- D e. RR   =>   |- (ph -> (0 < D -> E.z e. NN A.x e. NN A.y e. NN ((z <_ x /\ z <_ y) -> (normh` ((F` x) -h (F` y))) < D)))
Theorem	projlem29 9214	Part of Lemma 3.6 of [Beran] p. 101: 'Hence, {yn} is a Cauchy sequence.' Used by projlem30 9215.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))   =>   |- (ph -> F e. Cauchy)
Theorem	projlem30 9215	Part of Lemma 3.6 of [Beran] p. 101: 'By completeness of W, there exists y0 e. W such that yn->y0.' Used by projlem31 9216.
|- A e. H~   &   |- H e. CH   &   |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}   &   |- R = -usup(S, RR, < )   &   |- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (normh