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| Mirrors > Home > HSE Home > Th. List > idunop | Structured version Visualization version GIF version | ||
| Description: The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| idunop | ⊢ ( I ↾ ℋ) ∈ UniOp | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | f1oi 6885 | . . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ | |
| 2 | f1ofo 6854 | . . 3 ⊢ (( I ↾ ℋ): ℋ–1-1-onto→ ℋ → ( I ↾ ℋ): ℋ–onto→ ℋ) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ ( I ↾ ℋ): ℋ–onto→ ℋ | 
| 4 | fvresi 7194 | . . . 4 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥) | |
| 5 | fvresi 7194 | . . . 4 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦) | |
| 6 | 4, 5 | oveqan12d 7451 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)) | 
| 7 | 6 | rgen2 3198 | . 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦) | 
| 8 | elunop 31892 | . 2 ⊢ (( I ↾ ℋ) ∈ UniOp ↔ (( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦))) | |
| 9 | 3, 7, 8 | mpbir2an 711 | 1 ⊢ ( I ↾ ℋ) ∈ UniOp | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 ∀wral 3060 I cid 5576 ↾ cres 5686 –onto→wfo 6558 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 ℋchba 30939 ·ih csp 30942 UniOpcuo 30969 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-hilex 31019 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 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-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-unop 31863 | 
| This theorem is referenced by: idlnop 32012 | 
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