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Theorem unop 30178

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 30135	. . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))
2	1	simprbi 496	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
3	2	3ad2ant1 1131	. 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
4		fveq2 6756	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	4	oveq1d 7270	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)))
6		oveq1 7262	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih 𝑦) = (𝐴 ·_ih 𝑦))
7	5, 6	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦) ↔ ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih 𝑦)))
8		fveq2 6756	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
9	8	oveq2d 7271	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)))
10		oveq2 7263	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih 𝑦) = (𝐴 ·_ih 𝐵))
11	9, 10	eqeq12d 2754	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih 𝑦) ↔ ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)) = (𝐴 ·_ih 𝐵)))
12	7, 11	rspc2v 3562	. . 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 1128	. 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 15	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 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 –onto→wfo 6416 ‘cfv 6418 (class class class)co 7255 ℋchba 29182 ·_ih csp 29185 UniOpcuo 29212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-hilex 29262

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-unop 30106

This theorem is referenced by: unopf1o 30179 unopnorm 30180 cnvunop 30181 unopadj 30182 counop 30184

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