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Mirrors > Home > HSE Home > Th. List > cnvunop | Structured version Visualization version GIF version |
Description: The inverse (converse) of a unitary operator in Hilbert space is unitary. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnvunop | ⊢ (𝑇 ∈ UniOp → ◡𝑇 ∈ UniOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unopf1o 31945 | . . 3 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ) | |
2 | f1ocnv 6861 | . . . 4 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → ◡𝑇: ℋ–1-1-onto→ ℋ) | |
3 | f1ofo 6856 | . . . 4 ⊢ (◡𝑇: ℋ–1-1-onto→ ℋ → ◡𝑇: ℋ–onto→ ℋ) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → ◡𝑇: ℋ–onto→ ℋ) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝑇 ∈ UniOp → ◡𝑇: ℋ–onto→ ℋ) |
6 | simpl 482 | . . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ UniOp) | |
7 | fof 6821 | . . . . . . . 8 ⊢ (◡𝑇: ℋ–onto→ ℋ → ◡𝑇: ℋ⟶ ℋ) | |
8 | 5, 7 | syl 17 | . . . . . . 7 ⊢ (𝑇 ∈ UniOp → ◡𝑇: ℋ⟶ ℋ) |
9 | 8 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ) → (◡𝑇‘𝑥) ∈ ℋ) |
10 | 9 | adantrr 717 | . . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (◡𝑇‘𝑥) ∈ ℋ) |
11 | 8 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (◡𝑇‘𝑦) ∈ ℋ) |
12 | 11 | adantrl 716 | . . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (◡𝑇‘𝑦) ∈ ℋ) |
13 | unop 31944 | . . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ (◡𝑇‘𝑥) ∈ ℋ ∧ (◡𝑇‘𝑦) ∈ ℋ) → ((𝑇‘(◡𝑇‘𝑥)) ·ih (𝑇‘(◡𝑇‘𝑦))) = ((◡𝑇‘𝑥) ·ih (◡𝑇‘𝑦))) | |
14 | 6, 10, 12, 13 | syl3anc 1370 | . . . 4 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘(◡𝑇‘𝑥)) ·ih (𝑇‘(◡𝑇‘𝑦))) = ((◡𝑇‘𝑥) ·ih (◡𝑇‘𝑦))) |
15 | f1ocnvfv2 7297 | . . . . . . 7 ⊢ ((𝑇: ℋ–1-1-onto→ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘(◡𝑇‘𝑥)) = 𝑥) | |
16 | 15 | adantrr 717 | . . . . . 6 ⊢ ((𝑇: ℋ–1-1-onto→ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘(◡𝑇‘𝑥)) = 𝑥) |
17 | f1ocnvfv2 7297 | . . . . . . 7 ⊢ ((𝑇: ℋ–1-1-onto→ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘(◡𝑇‘𝑦)) = 𝑦) | |
18 | 17 | adantrl 716 | . . . . . 6 ⊢ ((𝑇: ℋ–1-1-onto→ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘(◡𝑇‘𝑦)) = 𝑦) |
19 | 16, 18 | oveq12d 7449 | . . . . 5 ⊢ ((𝑇: ℋ–1-1-onto→ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘(◡𝑇‘𝑥)) ·ih (𝑇‘(◡𝑇‘𝑦))) = (𝑥 ·ih 𝑦)) |
20 | 1, 19 | sylan 580 | . . . 4 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘(◡𝑇‘𝑥)) ·ih (𝑇‘(◡𝑇‘𝑦))) = (𝑥 ·ih 𝑦)) |
21 | 14, 20 | eqtr3d 2777 | . . 3 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((◡𝑇‘𝑥) ·ih (◡𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) |
22 | 21 | ralrimivva 3200 | . 2 ⊢ (𝑇 ∈ UniOp → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((◡𝑇‘𝑥) ·ih (◡𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) |
23 | elunop 31901 | . 2 ⊢ (◡𝑇 ∈ UniOp ↔ (◡𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((◡𝑇‘𝑥) ·ih (◡𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) | |
24 | 5, 22, 23 | sylanbrc 583 | 1 ⊢ (𝑇 ∈ UniOp → ◡𝑇 ∈ UniOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ◡ccnv 5688 ⟶wf 6559 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ℋchba 30948 ·ih csp 30951 UniOpcuo 30978 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-hilex 31028 ax-hfvadd 31029 ax-hvcom 31030 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvdistr2 31038 ax-hvmul0 31039 ax-hfi 31108 ax-his1 31111 ax-his2 31112 ax-his3 31113 ax-his4 31114 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137 df-hvsub 31000 df-unop 31872 |
This theorem is referenced by: unoplin 31949 unopadj2 31967 |
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