hmop - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > hmop		Structured version Visualization version GIF version

Theorem hmop 32557

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 32508	. . . 4 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
2	1	simprbi 503	. . 3 ⊢ (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
3	2	3ad2ant1 1151	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
4		oveq1 7453	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝑦)))
5		fveq2 6911	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
6	5	oveq1d 7461	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·_ih 𝑦) = ((𝑇‘𝐴) ·_ih 𝑦))
7	4, 6	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) ↔ (𝐴 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih 𝑦)))
8		fveq2 6911	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
9	8	oveq2d 7462	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih (𝑇‘𝐵)))
10		oveq2 7454	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·_ih 𝑦) = ((𝑇‘𝐴) ·_ih 𝐵))
11	9, 10	eqeq12d 2779	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih 𝑦) ↔ (𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵)))
12	7, 11	rspc2v 3590	. . 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 1148	. 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 16	1 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·_ih 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ⟶wf 6557 ‘cfv 6561 (class class class)co 7446 ℋchba 31554 ·_ih csp 31557 HrmOpcho 31585

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-hilex 31634

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7449 df-oprab 7450 df-mpo 7451 df-map 8880 df-hmop 32479

This theorem is used by: hmopre 32558 hmopadj 32574 hmoplin 32577 eighmre 32598 eighmorth 32599 hmopbdoptHIL 32623 hmops 32655 hmopm 32656 hmopco 32658 leopsq 32764 hmopidmpji 32787

W3C validator