Theorem projlem21 9206

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 9202). Here we just show the sequence value belongs to the closed subspace H. Used by projlem27 9212 projlem28 9213.

Hypothesis

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

Assertion

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

Distinct variable groups:   w,A   w,D   w,F   w,R

Proof of Theorem projlem21

Step	Hyp	Ref	Expression
1		ffvelrn 3814	. . 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)) < (normh` ((F` w) -h A)) /\ (normh` ((F` w) -h A)) < (R + (1 / w)))))
3	2	pm3.26bi 322	. . 3 |- (ph -> F:NN-->H)
4	1, 3	sylan 448	. 2 |- ((ph /\ D e. NN) -> (F` D) e. H)
5	4	ex 373	1 |- (ph -> (D e. NN -> (F` D) e. H))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958  A.wral 1645   class class class wbr 2619  -->wf 3178  ` cfv 3182  (class class class)co 3963  1c1 5235   + caddc 5237   - cmin 5292   / cdiv 5294  NNcn 5296   < clt 5486   -h cmv 8792  normhcno 8794

This theorem is referenced by:  projlem27 9212  projlem28 9213

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198

Copyright terms: Public domain