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Theorem unop 28744
Description: Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
unop ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵))

Proof of Theorem unop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elunop 28701 . . . 4 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
21simprbi 480 . . 3 (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))
323ad2ant1 1080 . 2 ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))
4 fveq2 6178 . . . . . 6 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
54oveq1d 6650 . . . . 5 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih (𝑇𝑦)))
6 oveq1 6642 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih 𝑦) = (𝐴 ·ih 𝑦))
75, 6eqeq12d 2635 . . . 4 (𝑥 = 𝐴 → (((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇𝐴) ·ih (𝑇𝑦)) = (𝐴 ·ih 𝑦)))
8 fveq2 6178 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
98oveq2d 6651 . . . . 5 (𝑦 = 𝐵 → ((𝑇𝐴) ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih (𝑇𝐵)))
10 oveq2 6643 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih 𝑦) = (𝐴 ·ih 𝐵))
119, 10eqeq12d 2635 . . . 4 (𝑦 = 𝐵 → (((𝑇𝐴) ·ih (𝑇𝑦)) = (𝐴 ·ih 𝑦) ↔ ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)))
127, 11rspc2v 3317 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)))
13123adant1 1077 . 2 ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)))
143, 13mpd 15 1 ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  wral 2909  ontowfo 5874  cfv 5876  (class class class)co 6635  chil 27746   ·ih csp 27749  UniOpcuo 27776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-hilex 27826
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-unop 28672
This theorem is referenced by:  unopf1o  28745  unopnorm  28746  cnvunop  28747  unopadj  28748  counop  28750
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