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Theorem hmop 29712
 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 29663 . . . 4 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
21simprbi 500 . . 3 (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
323ad2ant1 1130 . 2 ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
4 oveq1 7142 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
5 fveq2 6645 . . . . . 6 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
65oveq1d 7150 . . . . 5 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
74, 6eqeq12d 2814 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦)))
8 fveq2 6645 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
98oveq2d 7151 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
10 oveq2 7143 . . . . 5 (𝑦 = 𝐵 → ((𝑇𝐴) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝐵))
119, 10eqeq12d 2814 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
127, 11rspc2v 3581 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
13123adant1 1127 . 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 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ℋchba 28709   ·ih csp 28712  HrmOpcho 28740 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-hilex 28789 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8393  df-hmop 29634 This theorem is referenced by:  hmopre  29713  hmopadj  29729  hmoplin  29732  eighmre  29753  eighmorth  29754  hmopbdoptHIL  29778  hmops  29810  hmopm  29811  hmopco  29813  leopsq  29919  hmopidmpji  29942
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