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Theorem hmop 31951
Description: Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmop ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵))

Proof of Theorem hmop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elhmop 31902 . . . 4 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
21simprbi 496 . . 3 (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
323ad2ant1 1132 . 2 ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
4 oveq1 7438 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
5 fveq2 6907 . . . . . 6 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
65oveq1d 7446 . . . . 5 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
74, 6eqeq12d 2751 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦)))
8 fveq2 6907 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
98oveq2d 7447 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
10 oveq2 7439 . . . . 5 (𝑦 = 𝐵 → ((𝑇𝐴) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝐵))
119, 10eqeq12d 2751 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
127, 11rspc2v 3633 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
13123adant1 1129 . 2 ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
143, 13mpd 15 1 ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wf 6559  cfv 6563  (class class class)co 7431  chba 30948   ·ih csp 30951  HrmOpcho 30979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-hmop 31873
This theorem is referenced by:  hmopre  31952  hmopadj  31968  hmoplin  31971  eighmre  31992  eighmorth  31993  hmopbdoptHIL  32017  hmops  32049  hmopm  32050  hmopco  32052  leopsq  32158  hmopidmpji  32181
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