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Mirrors > Home > HSE Home > Th. List > unopadj2 | Structured version Visualization version GIF version |
Description: The adjoint of a unitary operator is its inverse (converse). Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 23-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
unopadj2 | ⊢ (𝑇 ∈ UniOp → (adjℎ‘𝑇) = ◡𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unoplin 31929 | . . 3 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp) | |
2 | lnopf 31868 | . . 3 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ) |
4 | cnvunop 31927 | . . 3 ⊢ (𝑇 ∈ UniOp → ◡𝑇 ∈ UniOp) | |
5 | unoplin 31929 | . . 3 ⊢ (◡𝑇 ∈ UniOp → ◡𝑇 ∈ LinOp) | |
6 | lnopf 31868 | . . 3 ⊢ (◡𝑇 ∈ LinOp → ◡𝑇: ℋ⟶ ℋ) | |
7 | 4, 5, 6 | 3syl 18 | . 2 ⊢ (𝑇 ∈ UniOp → ◡𝑇: ℋ⟶ ℋ) |
8 | unopadj 31928 | . . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (◡𝑇‘𝑦))) | |
9 | 8 | 3expib 1123 | . . 3 ⊢ (𝑇 ∈ UniOp → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (◡𝑇‘𝑦)))) |
10 | 9 | ralrimivv 3199 | . 2 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (◡𝑇‘𝑦))) |
11 | adjeq 31944 | . 2 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ◡𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (◡𝑇‘𝑦))) → (adjℎ‘𝑇) = ◡𝑇) | |
12 | 3, 7, 10, 11 | syl3anc 1373 | 1 ⊢ (𝑇 ∈ UniOp → (adjℎ‘𝑇) = ◡𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3060 ◡ccnv 5682 ⟶wf 6555 ‘cfv 6559 (class class class)co 7429 ℋchba 30928 ·ih csp 30931 LinOpclo 30956 UniOpcuo 30958 adjℎcado 30964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-hilex 31008 ax-hfvadd 31009 ax-hvcom 31010 ax-hvass 31011 ax-hv0cl 31012 ax-hvaddid 31013 ax-hfvmul 31014 ax-hvmulid 31015 ax-hvdistr2 31018 ax-hvmul0 31019 ax-hfi 31088 ax-his1 31091 ax-his2 31092 ax-his3 31093 ax-his4 31094 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-po 5590 df-so 5591 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-er 8741 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-2 12325 df-cj 15134 df-re 15135 df-im 15136 df-hvsub 30980 df-lnop 31850 df-unop 31852 df-adjh 31858 |
This theorem is referenced by: (None) |
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