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Theorem unop 32574

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 32531	. . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ⟶onto ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))
2	1	simprbi 503	. . 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 6916	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	4	oveq1d 7466	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)))
6		oveq1 7458	. . . . 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 6916	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
9	8	oveq2d 7467	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)))
10		oveq2 7459	. . . . 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 3590	. . 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

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ⟶ontowfo 6564 ‘cfv 6566 (class class class)co 7451 ℋchba 31578 ·_ih csp 31581 UniOpcuo 31608

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-rep 5238 ax-sep 5255 ax-nul 5266 ax-pr 5398 ax-hilex 31658

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-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-ov 7454 df-unop 32502

This theorem is used by: unopf1o 32575 unopnorm 32576 cnvunop 32577 unopadj 32578 counop 32580

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