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Theorem unop 29114
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 29071 . . . 4 (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))
21simprbi 484 . . 3 (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))
323ad2ant1 1127 . 2 ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦))
4 fveq2 6332 . . . . . 6 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
54oveq1d 6808 . . . . 5 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih (𝑇𝑦)))
6 oveq1 6800 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih 𝑦) = (𝐴 ·ih 𝑦))
75, 6eqeq12d 2786 . . . 4 (𝑥 = 𝐴 → (((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇𝐴) ·ih (𝑇𝑦)) = (𝐴 ·ih 𝑦)))
8 fveq2 6332 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
98oveq2d 6809 . . . . 5 (𝑦 = 𝐵 → ((𝑇𝐴) ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih (𝑇𝐵)))
10 oveq2 6801 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih 𝑦) = (𝐴 ·ih 𝐵))
119, 10eqeq12d 2786 . . . 4 (𝑦 = 𝐵 → (((𝑇𝐴) ·ih (𝑇𝑦)) = (𝐴 ·ih 𝑦) ↔ ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)))
127, 11rspc2v 3472 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵)))
13123adant1 1124 . 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 1071   = wceq 1631  wcel 2145  wral 3061  ontowfo 6029  cfv 6031  (class class class)co 6793  chil 28116   ·ih csp 28119  UniOpcuo 28146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-hilex 28196
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-unop 29042
This theorem is referenced by:  unopf1o  29115  unopnorm  29116  cnvunop  29117  unopadj  29118  counop  29120
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