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Mirrors > Home > HSE Home > Th. List > unopadj | Structured version Visualization version GIF version |
Description: The inverse (converse) of a unitary operator is its adjoint. Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
unopadj | ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih (◡𝑇‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unopf1o 29692 | . . . . 5 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ) | |
2 | f1ocnvfv2 7033 | . . . . 5 ⊢ ((𝑇: ℋ–1-1-onto→ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(◡𝑇‘𝐵)) = 𝐵) | |
3 | 1, 2 | sylan 582 | . . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝐵 ∈ ℋ) → (𝑇‘(◡𝑇‘𝐵)) = 𝐵) |
4 | 3 | 3adant2 1127 | . . 3 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(◡𝑇‘𝐵)) = 𝐵) |
5 | 4 | oveq2d 7171 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih (𝑇‘(◡𝑇‘𝐵))) = ((𝑇‘𝐴) ·ih 𝐵)) |
6 | f1ocnv 6626 | . . . . . 6 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → ◡𝑇: ℋ–1-1-onto→ ℋ) | |
7 | f1of 6614 | . . . . . 6 ⊢ (◡𝑇: ℋ–1-1-onto→ ℋ → ◡𝑇: ℋ⟶ ℋ) | |
8 | 1, 6, 7 | 3syl 18 | . . . . 5 ⊢ (𝑇 ∈ UniOp → ◡𝑇: ℋ⟶ ℋ) |
9 | 8 | ffvelrnda 6850 | . . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝐵 ∈ ℋ) → (◡𝑇‘𝐵) ∈ ℋ) |
10 | 9 | 3adant2 1127 | . . 3 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (◡𝑇‘𝐵) ∈ ℋ) |
11 | unop 29691 | . . 3 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ (◡𝑇‘𝐵) ∈ ℋ) → ((𝑇‘𝐴) ·ih (𝑇‘(◡𝑇‘𝐵))) = (𝐴 ·ih (◡𝑇‘𝐵))) | |
12 | 10, 11 | syld3an3 1405 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih (𝑇‘(◡𝑇‘𝐵))) = (𝐴 ·ih (◡𝑇‘𝐵))) |
13 | 5, 12 | eqtr3d 2858 | 1 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih (◡𝑇‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ◡ccnv 5553 ⟶wf 6350 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 ℋchba 28695 ·ih csp 28698 UniOpcuo 28725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-hilex 28775 ax-hfvadd 28776 ax-hvcom 28777 ax-hvass 28778 ax-hv0cl 28779 ax-hvaddid 28780 ax-hfvmul 28781 ax-hvmulid 28782 ax-hvdistr2 28785 ax-hvmul0 28786 ax-hfi 28855 ax-his1 28858 ax-his2 28859 ax-his3 28860 ax-his4 28861 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-2 11699 df-cj 14457 df-re 14458 df-im 14459 df-hvsub 28747 df-unop 29619 |
This theorem is referenced by: unoplin 29696 unopadj2 29714 |
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