Theorem cnlnadjlem1 10000

Description: Lemma for cnlnadj 10009 (Theorem 3.10 of [Beran] p. 104: every continuous linear operator has an adjoint). The value of the auxiliary functional G.

Hypotheses

Ref	Expression
cnlnadjlem.1	|- T e. LinOp
cnlnadjlem.2	|- T e. ConOp
cnlnadjlem.3	|- G = {<.g, h>. | (g e. H~ /\ h = ((T` g) .ih y))}

Assertion

Ref	Expression
cnlnadjlem1	|- (A e. H~ -> (G` A) = ((T` A) .ih y))

Distinct variable groups:   g,h,y,A   T,g,h,y

Proof of Theorem cnlnadjlem1

Step	Hyp	Ref	Expression
1		fveq2 3724	. . 3 |- (g = A -> (T` g) = (T` A))
2	1	opreq1d 3975	. 2 |- (g = A -> ((T` g) .ih y) = ((T` A) .ih y))
3		cnlnadjlem.3	. 2 |- G = {<.g, h>. | (g e. H~ /\ h = ((T` g) .ih y))}
4		oprex 3983	. 2 |- ((T` A) .ih y) e. V
5	2, 3, 4	fvopab4 3780	1 |- (A e. H~ -> (G` A) = ((T` A) .ih y))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {copab 2666  ` cfv 3182  (class class class)co 3963  H~chil 8788   .ih csp 8793  ConOpcco 8815  LinOpclo 8816

This theorem is referenced by:  cnlnadjlem2 10001  cnlnadjlem3 10002  cnlnadjlem4 10003  cnlnadjlem5 10004

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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