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Mirrors > Home > HSE Home > Th. List > unop | Structured version Visualization version GIF version |
Description: Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
unop | ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elunop 30988 | . . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) | |
2 | 1 | simprbi 497 | . . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) |
3 | 2 | 3ad2ant1 1133 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) |
4 | fveq2 6878 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 4 | oveq1d 7408 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih (𝑇‘𝑦))) |
6 | oveq1 7400 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih 𝑦) = (𝐴 ·ih 𝑦)) | |
7 | 5, 6 | eqeq12d 2747 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇‘𝐴) ·ih (𝑇‘𝑦)) = (𝐴 ·ih 𝑦))) |
8 | fveq2 6878 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵)) | |
9 | 8 | oveq2d 7409 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih (𝑇‘𝐵))) |
10 | oveq2 7401 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih 𝑦) = (𝐴 ·ih 𝐵)) | |
11 | 9, 10 | eqeq12d 2747 | . . . 4 ⊢ (𝑦 = 𝐵 → (((𝑇‘𝐴) ·ih (𝑇‘𝑦)) = (𝐴 ·ih 𝑦) ↔ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵))) |
12 | 7, 11 | rspc2v 3618 | . . 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 1130 | . 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 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 –onto→wfo 6530 ‘cfv 6532 (class class class)co 7393 ℋchba 30035 ·ih csp 30038 UniOpcuo 30065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-hilex 30115 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-unop 30959 |
This theorem is referenced by: unopf1o 31032 unopnorm 31033 cnvunop 31034 unopadj 31035 counop 31037 |
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