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Theorem unop 32267

Description: Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
unop	⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)) = (𝐴 ·_ih 𝐵))

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

Step	Hyp	Ref	Expression
1		elunop 32224	. . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))
2	1	simprbi 502	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
3	2	3ad2ant1 1151	. 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
4		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	4	oveq1d 7425	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)))
6		oveq1 7417	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih 𝑦) = (𝐴 ·_ih 𝑦))
7	5, 6	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦) ↔ ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih 𝑦)))
8		fveq2 6881	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
9	8	oveq2d 7426	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)))
10		oveq2 7418	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih 𝑦) = (𝐴 ·_ih 𝐵))
11	9, 10	eqeq12d 2779	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih 𝑦) ↔ ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)) = (𝐴 ·_ih 𝐵)))
12	7, 11	rspc2v 3592	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦) → ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)) = (𝐴 ·_ih 𝐵)))
13	12	3adant1 1148	. 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦) → ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)) = (𝐴 ·_ih 𝐵)))
14	3, 13	mpd 16	1 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)) = (𝐴 ·_ih 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 –onto→wfo 6534 ‘cfv 6536 (class class class)co 7410 ℋchba 31271 ·_ih csp 31274 UniOpcuo 31301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pr 5404 ax-hilex 31351

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-unop 32195

This theorem is referenced by: unopf1o 32268 unopnorm 32269 cnvunop 32270 unopadj 32271 counop 32273

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