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Theorem hmop 30864
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 30815 . . . 4 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
21simprbi 497 . . 3 (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
323ad2ant1 1133 . 2 ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
4 oveq1 7364 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
5 fveq2 6842 . . . . . 6 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
65oveq1d 7372 . . . . 5 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
74, 6eqeq12d 2752 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦)))
8 fveq2 6842 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
98oveq2d 7373 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
10 oveq2 7365 . . . . 5 (𝑦 = 𝐵 → ((𝑇𝐴) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝐵))
119, 10eqeq12d 2752 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
127, 11rspc2v 3590 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
13123adant1 1130 . 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 1087   = wceq 1541  wcel 2106  wral 3064  wf 6492  cfv 6496  (class class class)co 7357  chba 29861   ·ih csp 29864  HrmOpcho 29892
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-hilex 29941
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-hmop 30786
This theorem is referenced by:  hmopre  30865  hmopadj  30881  hmoplin  30884  eighmre  30905  eighmorth  30906  hmopbdoptHIL  30930  hmops  30962  hmopm  30963  hmopco  30965  leopsq  31071  hmopidmpji  31094
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