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Mirrors > Home > HSE Home > Th. List > hmop | Structured version Visualization version GIF version |
Description: Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hmop | ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elhmop 29434 | . . . 4 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) | |
2 | 1 | simprbi 489 | . . 3 ⊢ (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) |
3 | 2 | 3ad2ant1 1113 | . 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) |
4 | oveq1 6985 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝑦))) | |
5 | fveq2 6501 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
6 | 5 | oveq1d 6993 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦)) |
7 | 4, 6 | eqeq12d 2793 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih 𝑦))) |
8 | fveq2 6501 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵)) | |
9 | 8 | oveq2d 6994 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝐵))) |
10 | oveq2 6986 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝐵)) | |
11 | 9, 10 | eqeq12d 2793 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵))) |
12 | 7, 11 | rspc2v 3548 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵))) |
13 | 12 | 3adant1 1110 | . 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵))) |
14 | 3, 13 | mpd 15 | 1 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ℋchba 28478 ·ih csp 28481 HrmOpcho 28509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-hilex 28558 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3684 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-id 5313 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-fv 6198 df-ov 6981 df-oprab 6982 df-mpo 6983 df-map 8210 df-hmop 29405 |
This theorem is referenced by: hmopre 29484 hmopadj 29500 hmoplin 29503 eighmre 29524 eighmorth 29525 hmopbdoptHIL 29549 hmops 29581 hmopm 29582 hmopco 29584 leopsq 29690 hmopidmpji 29713 |
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