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Theorem unop 32075

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 32032	. . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))
2	1	simprbi 501	. . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
3	2	3ad2ant1 1145	. 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
4		fveq2 6862	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	4	oveq1d 7406	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)))
6		oveq1 7398	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·_ih 𝑦) = (𝐴 ·_ih 𝑦))
7	5, 6	eqeq12d 2777	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦) ↔ ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih 𝑦)))
8		fveq2 6862	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵))
9	8	oveq2d 7407	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)))
10		oveq2 7399	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·_ih 𝑦) = (𝐴 ·_ih 𝐵))
11	9, 10	eqeq12d 2777	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝑇‘𝐴) ·_ih (𝑇‘𝑦)) = (𝐴 ·_ih 𝑦) ↔ ((𝑇‘𝐴) ·_ih (𝑇‘𝐵)) = (𝐴 ·_ih 𝐵)))
12	7, 11	rspc2v 3591	. . 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 1142	. 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 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 –onto→wfo 6514 ‘cfv 6516 (class class class)co 7391 ℋchba 31079 ·_ih csp 31082 UniOpcuo 31109

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pr 5387 ax-hilex 31159

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7394 df-unop 32003

This theorem is referenced by: unopf1o 32076 unopnorm 32077 cnvunop 32078 unopadj 32079 counop 32081

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