Theorem hommvalt 9469

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 8824	. . . 4 |- H~ e. V
2	1	opabex2 3606	. . 3 |- {<.x, y>. | (x e. H~ /\ y = (A .h (T` x)))} e. V
3		opreq1 3963	. . . . . 6 |- (f = A -> (f .h (g` x)) = (A .h (g` x)))
4	3	eqeq2d 1484	. . . . 5 |- (f = A -> (y = (f .h (g` x)) <-> y = (A .h (g` x))))
5	4	anbi2d 615	. . . 4 |- (f = A -> ((x e. H~ /\ y = (f .h (g` x))) <-> (x e. H~ /\ y = (A .h (g` x)))))
6	5	opabbidv 2666	. . 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 3718	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
8	7	opreq2d 3971	. . . . . 6 |- (g = T -> (A .h (g` x)) = (A .h (T` x)))
9	8	eqeq2d 1484	. . . . 5 |- (g = T -> (y = (A .h (g` x)) <-> y = (A .h (T` x))))
10	9	anbi2d 615	. . . 4 |- (g = T -> ((x e. H~ /\ y = (A .h (g` x))) <-> (x e. H~ /\ y = (A .h (T` x)))))
11	10	opabbidv 2666	. . 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 9464	. . . 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 4327	. . . . . . 7 |- (g e. (H~ ^m H~) <-> g:H~-->H~)
14	13	anbi2i 480	. . . . . 6 |- ((f e. CC /\ g e. (H~ ^m H~)) <-> (f e. CC /\ g:H~-->H~))
15	14	anbi1i 481	. . . . 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 3992	. . . 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 1496	. . 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 4023	. 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 4327	. 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 955   e. wcel 957  {copab 2662  -->wf 3174  ` cfv 3178  (class class class)co 3958  {copab2 3959   ^m cm 4315  CCcc 5215  H~chil 8743   .h csm 8745   .op chot 8763

This theorem is referenced by:  homvalt 9475  homulclt 9642

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-hilex 8824

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-opr 3960  df-oprab 3961  df-map 4317  df-homul 9464

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