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Theorem hfmmvalt 9432
Description: Value of the scalar product with a Hilbert space functional.
Assertion
Ref Expression
hfmmvalt |- ((A e. CC /\ T:H~-->CC) -> (A .fn T) = {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))})
Distinct variable groups:   x,y,A   x,T,y
Proof of Theorem hfmmvalt
StepHypRef Expression
1 ax-hilex 8790 . . . 4 |- H~ e. V
21opabex2 3596 . . 3 |- {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))} e. V
3 opreq1 3953 . . . . . 6 |- (f = A -> (f x. (g` x)) = (A x. (g` x)))
43eqeq2d 1478 . . . . 5 |- (f = A -> (y = (f x. (g` x)) <-> y = (A x. (g` x))))
54anbi2d 614 . . . 4 |- (f = A -> ((x e. H~ /\ y = (f x. (g` x))) <-> (x e. H~ /\ y = (A x. (g` x)))))
65opabbidv 2660 . . 3 |- (f = A -> {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))} = {<.x, y>. | (x e. H~ /\ y = (A x. (g` x)))})
7 fveq1 3708 . . . . . . 7 |- (g = T -> (g` x) = (T` x))
87opreq2d 3961 . . . . . 6 |- (g = T -> (A x. (g` x)) = (A x. (T` x)))
98eqeq2d 1478 . . . . 5 |- (g = T -> (y = (A x. (g` x)) <-> y = (A x. (T` x))))
109anbi2d 614 . . . 4 |- (g = T -> ((x e. H~ /\ y = (A x. (g` x))) <-> (x e. H~ /\ y = (A x. (T` x)))))
1110opabbidv 2660 . . 3 |- (g = T -> {<.x, y>. | (x e. H~ /\ y = (A x. (g` x)))} = {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))})
12 df-hfmul 9427 . . . 4 |- .fn = {<.<.f, g>., h>. | ((f e. CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))})}
13 axcnex 5239 . . . . . . . 8 |- CC e. V
1413, 1elmap 4318 . . . . . . 7 |- (g e. (CC ^m H~) <-> g:H~-->CC)
1514anbi2i 479 . . . . . 6 |- ((f e. CC /\ g e. (CC ^m H~)) <-> (f e. CC /\ g:H~-->CC))
1615anbi1i 480 . . . . 5 |- (((f e. CC /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))}) <-> ((f e. CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))}))
1716oprabbii 3982 . . . 4 |- {<.<.f, g>., h>. | ((f e. CC /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))})} = {<.<.f, g>., h>. | ((f e. CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))})}
1812, 17eqtr4 1490 . . 3 |- .fn = {<.<.f, g>., h>. | ((f e. CC /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))})}
192, 6, 11, 18oprabval2 4013 . 2 |- ((A e. CC /\ T e. (CC ^m H~)) -> (A .fn T) = {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))})
2013, 1elmap 4318 . 2 |- (T e. (CC ^m H~) <-> T:H~-->CC)
2119, 20sylan2br 453 1 |- ((A e. CC /\ T:H~-->CC) -> (A .fn T) = {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))})
Colors of variables: wff set class
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   x. cmul 5211  H~chil 8727   .fn chft 8750
This theorem is referenced by:  hfmvalt 9439  brafnmult 9791  kbass2t 9962
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-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hilex 8790
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  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-ral 1641  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-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  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-qs 4250  df-map 4308  df-ni 4972  df-nq 5010  df-np 5058  df-nr 5139  df-c 5212  df-hfmul 9427
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