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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 29641 | . . . 4 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) | |
2 | 1 | simprbi 499 | . . 3 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) |
3 | 2 | 3ad2ant1 1128 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) |
4 | fveq2 6663 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
5 | 4 | oveq1d 7163 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih (𝑇‘𝑦))) |
6 | oveq1 7155 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih 𝑦) = (𝐴 ·ih 𝑦)) | |
7 | 5, 6 | eqeq12d 2835 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇‘𝐴) ·ih (𝑇‘𝑦)) = (𝐴 ·ih 𝑦))) |
8 | fveq2 6663 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵)) | |
9 | 8 | oveq2d 7164 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih (𝑇‘𝐵))) |
10 | oveq2 7156 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih 𝑦) = (𝐴 ·ih 𝐵)) | |
11 | 9, 10 | eqeq12d 2835 | . . . 4 ⊢ (𝑦 = 𝐵 → (((𝑇‘𝐴) ·ih (𝑇‘𝑦)) = (𝐴 ·ih 𝑦) ↔ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵))) |
12 | 7, 11 | rspc2v 3631 | . . 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 1125 | . 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 1082 = wceq 1531 ∈ wcel 2108 ∀wral 3136 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7148 ℋchba 28688 ·ih csp 28691 UniOpcuo 28718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-hilex 28768 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-unop 29612 |
This theorem is referenced by: unopf1o 29685 unopnorm 29686 cnvunop 29687 unopadj 29688 counop 29690 |
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