Theorem hommvalt 9429

Description: Value of the scalar product with a Hilbert space operator.

Assertion

Ref	Expression
hommvalt	|- ((A e. CC /\ T:H~-->H~) -> (A .op T) = {<.x, y>. | (x e. H~ /\ y = (A .h (T` x)))})

Distinct variable groups:   x,y,A   x,T,y

Proof of Theorem hommvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8790	. . . 4 |- H~ e. V
2	1	opabex2 3596	. . 3 |- {<.x, y>. | (x e. H~ /\ y = (A .h (T` x)))} e. V
3		opreq1 3953	. . . . . 6 |- (f = A -> (f .h (g` x)) = (A .h (g` x)))
4	3	eqeq2d 1478	. . . . 5 |- (f = A -> (y = (f .h (g` x)) <-> y = (A .h (g` x))))
5	4	anbi2d 614	. . . 4 |- (f = A -> ((x e. H~ /\ y = (f .h (g` x))) <-> (x e. H~ /\ y = (A .h (g` x)))))
6	5	opabbidv 2660	. . 3 |- (f = A -> {<.x, y>. | (x e. H~ /\ y = (f .h (g` x)))} = {<.x, y>. | (x e. H~ /\ y = (A .h (g` x)))})
7		fveq1 3708	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
8	7	opreq2d 3961	. . . . . 6 |- (g = T -> (A .h (g` x)) = (A .h (T` x)))
9	8	eqeq2d 1478	. . . . 5 |- (g = T -> (y = (A .h (g` x)) <-> y = (A .h (T` x))))
10	9	anbi2d 614	. . . 4 |- (g = T -> ((x e. H~ /\ y = (A .h (g` x))) <-> (x e. H~ /\ y = (A .h (T` x)))))
11	10	opabbidv 2660	. . 3 |- (g = T -> {<.x, y>. | (x e. H~ /\ y = (A .h (g` x)))} = {<.x, y>. | (x e. H~ /\ y = (A .h (T` x)))})
12		df-homul 9424	. . . 4 |- .op = {<.<.f, g>., h>. | ((f e. CC /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = (f .h (g` x)))})}
13	1, 1	elmap 4318	. . . . . . 7 |- (g e. (H~ ^m H~) <-> g:H~-->H~)
14	13	anbi2i 479	. . . . . 6 |- ((f e. CC /\ g e. (H~ ^m H~)) <-> (f e. CC /\ g:H~-->H~))
15	14	anbi1i 480	. . . . 5 |- (((f e. CC /\ g e. (H~ ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f .h (g` x)))}) <-> ((f e. CC /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = (f .h (g` x)))}))
16	15	oprabbii 3982	. . . 4 |- {<.<.f, g>., h>. | ((f e. CC /\ g e. (H~ ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f .h (g` x)))})} = {<.<.f, g>., h>. | ((f e. CC /\ g:H~-->H~) /\ h = {<.x, y>. | (x e. H~ /\ y = (f .h (g` x)))})}
17	12, 16	eqtr4 1490	. . 3 |- .op = {<.<.f, g>., h>. | ((f e. CC /\ g e. (H~ ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f .h (g` x)))})}
18	2, 6, 11, 17	oprabval2 4013	. 2 |- ((A e. CC /\ T e. (H~ ^m H~)) -> (A .op T) = {<.x, y>. | (x e. H~ /\ y = (A .h (T` x)))})
19	1, 1	elmap 4318	. 2 |- (T e. (H~ ^m H~) <-> T:H~-->H~)
20	18, 19	sylan2br 453	1 |- ((A e. CC /\ T:H~-->H~) -> (A .op T) = {<.x, y>. | (x e. H~ /\ y = (A .h (T` x)))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  {copab 2656  -->wf 3168  ` cfv 3172  (class class class)co 3948  {copab2 3949   ^m cm 4306  CCcc 5204  H~chil 8727   .h csm 8729   .op chot 8747

This theorem is referenced by:  homvalt 9435  homulclt 9602

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-oprab 3951  df-map 4308  df-homul 9424

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