![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 29334 | . . 3 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp) | |
2 | lnopf 29273 | . . 3 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ) |
4 | cnvunop 29332 | . . 3 ⊢ (𝑇 ∈ UniOp → ◡𝑇 ∈ UniOp) | |
5 | unoplin 29334 | . . 3 ⊢ (◡𝑇 ∈ UniOp → ◡𝑇 ∈ LinOp) | |
6 | lnopf 29273 | . . 3 ⊢ (◡𝑇 ∈ LinOp → ◡𝑇: ℋ⟶ ℋ) | |
7 | 4, 5, 6 | 3syl 18 | . 2 ⊢ (𝑇 ∈ UniOp → ◡𝑇: ℋ⟶ ℋ) |
8 | unopadj 29333 | . . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (◡𝑇‘𝑦))) | |
9 | 8 | 3expib 1158 | . . 3 ⊢ (𝑇 ∈ UniOp → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (◡𝑇‘𝑦)))) |
10 | 9 | ralrimivv 3179 | . 2 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (◡𝑇‘𝑦))) |
11 | adjeq 29349 | . 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 1496 | 1 ⊢ (𝑇 ∈ UniOp → (adjℎ‘𝑇) = ◡𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ◡ccnv 5341 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ℋchba 28331 ·ih csp 28334 LinOpclo 28359 UniOpcuo 28361 adjℎcado 28367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-hilex 28411 ax-hfvadd 28412 ax-hvcom 28413 ax-hvass 28414 ax-hv0cl 28415 ax-hvaddid 28416 ax-hfvmul 28417 ax-hvmulid 28418 ax-hvdistr2 28421 ax-hvmul0 28422 ax-hfi 28491 ax-his1 28494 ax-his2 28495 ax-his3 28496 ax-his4 28497 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-2 11414 df-cj 14216 df-re 14217 df-im 14218 df-hvsub 28383 df-lnop 29255 df-unop 29257 df-adjh 29263 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |