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Theorem hfmmval 31416
Description: Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hfmmval ((𝐴 ∈ β„‚ ∧ 𝑇: β„‹βŸΆβ„‚) β†’ (𝐴 Β·fn 𝑇) = (π‘₯ ∈ β„‹ ↦ (𝐴 Β· (π‘‡β€˜π‘₯))))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑇
Proof of Theorem hfmmval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 11186 . . 3 β„‚ ∈ V
2 ax-hilex 30676 . . 3 β„‹ ∈ V
31, 2elmap 8860 . 2 (𝑇 ∈ (β„‚ ↑m β„‹) ↔ 𝑇: β„‹βŸΆβ„‚)
4 oveq1 7408 . . . 4 (𝑓 = 𝐴 β†’ (𝑓 Β· (π‘”β€˜π‘₯)) = (𝐴 Β· (π‘”β€˜π‘₯)))
54mpteq2dv 5240 . . 3 (𝑓 = 𝐴 β†’ (π‘₯ ∈ β„‹ ↦ (𝑓 Β· (π‘”β€˜π‘₯))) = (π‘₯ ∈ β„‹ ↦ (𝐴 Β· (π‘”β€˜π‘₯))))
6 fveq1 6880 . . . . 5 (𝑔 = 𝑇 β†’ (π‘”β€˜π‘₯) = (π‘‡β€˜π‘₯))
76oveq2d 7417 . . . 4 (𝑔 = 𝑇 β†’ (𝐴 Β· (π‘”β€˜π‘₯)) = (𝐴 Β· (π‘‡β€˜π‘₯)))
87mpteq2dv 5240 . . 3 (𝑔 = 𝑇 β†’ (π‘₯ ∈ β„‹ ↦ (𝐴 Β· (π‘”β€˜π‘₯))) = (π‘₯ ∈ β„‹ ↦ (𝐴 Β· (π‘‡β€˜π‘₯))))
9 df-hfmul 31411 . . 3 Β·fn = (𝑓 ∈ β„‚, 𝑔 ∈ (β„‚ ↑m β„‹) ↦ (π‘₯ ∈ β„‹ ↦ (𝑓 Β· (π‘”β€˜π‘₯))))
102mptex 7216 . . 3 (π‘₯ ∈ β„‹ ↦ (𝐴 Β· (π‘‡β€˜π‘₯))) ∈ V
115, 8, 9, 10ovmpo 7560 . 2 ((𝐴 ∈ β„‚ ∧ 𝑇 ∈ (β„‚ ↑m β„‹)) β†’ (𝐴 Β·fn 𝑇) = (π‘₯ ∈ β„‹ ↦ (𝐴 Β· (π‘‡β€˜π‘₯))))
123, 11sylan2br 594 1 ((𝐴 ∈ β„‚ ∧ 𝑇: β„‹βŸΆβ„‚) β†’ (𝐴 Β·fn 𝑇) = (π‘₯ ∈ β„‹ ↦ (𝐴 Β· (π‘‡β€˜π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5221  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401   ↑m cmap 8815  β„‚cc 11103   Β· cmul 11110   β„‹chba 30596   Β·fn chft 30619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-hilex 30676
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8817  df-hfmul 31411
This theorem is referenced by:  hfmval  31421  brafnmul  31628  kbass2  31794
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