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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 29954 | . . . 4 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) | |
2 | 1 | simprbi 500 | . . 3 ⊢ (𝑇 ∈ HrmOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) |
3 | 2 | 3ad2ant1 1135 | . 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦)) |
4 | oveq1 7220 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝑦))) | |
5 | fveq2 6717 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴)) | |
6 | 5 | oveq1d 7228 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑇‘𝑥) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝑦)) |
7 | 4, 6 | eqeq12d 2753 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih 𝑦))) |
8 | fveq2 6717 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑇‘𝑦) = (𝑇‘𝐵)) | |
9 | 8 | oveq2d 7229 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ih (𝑇‘𝑦)) = (𝐴 ·ih (𝑇‘𝐵))) |
10 | oveq2 7221 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑇‘𝐴) ·ih 𝑦) = ((𝑇‘𝐴) ·ih 𝐵)) | |
11 | 9, 10 | eqeq12d 2753 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇‘𝑦)) = ((𝑇‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵))) |
12 | 7, 11 | rspc2v 3547 | . . 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 1132 | . 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 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℋchba 29000 ·ih csp 29003 HrmOpcho 29031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-hilex 29080 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-map 8510 df-hmop 29925 |
This theorem is referenced by: hmopre 30004 hmopadj 30020 hmoplin 30023 eighmre 30044 eighmorth 30045 hmopbdoptHIL 30069 hmops 30101 hmopm 30102 hmopco 30104 leopsq 30210 hmopidmpji 30233 |
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