Theorem projlem23 9208

Description: 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.

Hypothesis

Ref	Expression
projlem23.1	|- G = {<.x, y>. | (x e. NN /\ y = (normh` ((F` x) -h A)))}

Assertion

Ref	Expression
projlem23	|- (D e. NN -> (G` D) = (normh` ((F` D) -h A)))

Distinct variable groups:   x,y,D   x,A,y   x,F,y

Proof of Theorem projlem23

Step	Hyp	Ref	Expression
1		fveq2 3724	. . . 4 |- (x = D -> (F` x) = (F` D))
2	1	opreq1d 3975	. . 3 |- (x = D -> ((F` x) -h A) = ((F` D) -h A))
3	2	fveq2d 3728	. 2 |- (x = D -> (normh` ((F` x) -h A)) = (normh` ((F` D) -h A)))
4		projlem23.1	. 2 |- G = {<.x, y>. | (x e. NN /\ y = (normh` ((F` x) -h A)))}
5		fvex 3732	. 2 |- (normh` ((F` D) -h A)) e. V
6	3, 4, 5	fvopab4 3780	1 |- (D e. NN -> (G` D) = (normh` ((F` D) -h A)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {copab 2666  ` cfv 3182  (class class class)co 3963  NNcn 5296   -h cmv 8792  normhcno 8794

This theorem is referenced by:  projlem25 9210  projlem26 9211

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-9 965  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-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965

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