Theorem projlem21 5213

Description: 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 5209). Here we just show the sequence value belongs to the closed subspace H. Used by projlem27 5219 projlem28 5220.

Hypothesis

Ref	Expression
projlem21.1	|- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (norm` ((F` w) -v A)) /\ (norm` ((F` w) -v A)) < (R + (1 / w)))))

Assertion

Ref	Expression
projlem21	|- (ph -> (D e. NN -> (F` D) e. H))

Distinct variable group(s):   w,A   w,D   w,F   w,R

Proof of Theorem projlem21

Step	Hyp	Ref	Expression
1		ffvrn 2890	. . 3 |- ((F:NN-->H /\ D e. NN) -> (F` D) e. H)
2		projlem21.1	. . . 4 |- (ph <-> (F:NN-->H /\ A.w e. NN ((R - (1 / w)) < (norm` ((F` w) -v A)) /\ (norm` ((F` w) -v A)) < (R + (1 / w)))))
3	2	pm3.26bd 259	. . 3 |- (ph -> F:NN-->H)
4	1, 3	sylan 343	. 2 |- ((ph /\ D e. NN) -> (F` D) e. H)
5	4	exp 291	1 |- (ph -> (D e. NN -> (F` D) e. H))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   e. wcel 1092  A.wral 1201   class class class wbr 2054  -->wf 2418  ` cfv 2422  (class class class)co 3001  1c1 4029   + caddc 4031   < clt 4033   - cmin 4089   / cdiv 4091  NNcn 4093   -v cmv 4962  normcno 4964

This theorem is referenced by:  projlem27 5219  projlem28 5220

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438

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