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Theorem hmop 31851

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 (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵))

Proof of Theorem hmop Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elhmop 31802	. . . 4 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
2	1	simprbi 496	. . 3 ⊢ (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
3	2	3ad2ant1 1133	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
4		oveq1 7394	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝑦)))
5		fveq2 6858	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
6	5	oveq1d 7402	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·_ih 𝑦) = ((𝑇‘𝐴) ·_ih 𝑦))
7	4, 6	eqeq12d 2745	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) ↔ (𝐴 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih 𝑦)))
8		fveq2 6858	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
9	8	oveq2d 7403	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝐵)))
10		oveq2 7395	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·_ih 𝑦) = ((𝑇‘𝐴) ·_ih 𝐵))
11	9, 10	eqeq12d 2745	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih 𝑦) ↔ (𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵)))
12	7, 11	rspc2v 3599	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) → (𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵)))
13	12	3adant1 1130	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) → (𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵)))
14	3, 13	mpd 15	1 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℋchba 30848 ·_ih csp 30851 HrmOpcho 30879

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-hilex 30928

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-map 8801 df-hmop 31773

This theorem is referenced by: hmopre 31852 hmopadj 31868 hmoplin 31871 eighmre 31892 eighmorth 31893 hmopbdoptHIL 31917 hmops 31949 hmopm 31950 hmopco 31952 leopsq 32058 hmopidmpji 32081

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