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Theorem hmop 28165
 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 28116 . . . 4 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
21simprbi 479 . . 3 (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
323ad2ant1 1075 . 2 ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
4 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
5 fveq2 6103 . . . . . 6 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
65oveq1d 6564 . . . . 5 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
74, 6eqeq12d 2625 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦)))
8 fveq2 6103 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
98oveq2d 6565 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
10 oveq2 6557 . . . . 5 (𝑦 = 𝐵 → ((𝑇𝐴) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝐵))
119, 10eqeq12d 2625 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
127, 11rspc2v 3293 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
13123adant1 1072 . 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 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ℋchil 27160   ·ih csp 27163  HrmOpcho 27191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-hmop 28087 This theorem is referenced by:  hmopre  28166  hmopadj  28182  hmoplin  28185  eighmre  28206  eighmorth  28207  hmopbdoptHIL  28231  hmops  28263  hmopm  28264  hmopco  28266  leopsq  28372  hmopidmpji  28395
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