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Theorem hmop 29483
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 29434 . . . 4 (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))
21simprbi 489 . . 3 (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
323ad2ant1 1113 . 2 ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦))
4 oveq1 6985 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
5 fveq2 6501 . . . . . 6 (𝑥 = 𝐴 → (𝑇𝑥) = (𝑇𝐴))
65oveq1d 6993 . . . . 5 (𝑥 = 𝐴 → ((𝑇𝑥) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝑦))
74, 6eqeq12d 2793 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦)))
8 fveq2 6501 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
98oveq2d 6994 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
10 oveq2 6986 . . . . 5 (𝑦 = 𝐵 → ((𝑇𝐴) ·ih 𝑦) = ((𝑇𝐴) ·ih 𝐵))
119, 10eqeq12d 2793 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = ((𝑇𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
127, 11rspc2v 3548 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵)))
13123adant1 1110 . 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 1068   = wceq 1507  wcel 2050  wral 3088  wf 6186  cfv 6190  (class class class)co 6978  chba 28478   ·ih csp 28481  HrmOpcho 28509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-hilex 28558
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-map 8210  df-hmop 29405
This theorem is referenced by:  hmopre  29484  hmopadj  29500  hmoplin  29503  eighmre  29524  eighmorth  29525  hmopbdoptHIL  29549  hmops  29581  hmopm  29582  hmopco  29584  leopsq  29690  hmopidmpji  29713
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