|   | Hilbert Space Explorer | < Previous  
      Next > Nearby theorems | |
| 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 6886 | . . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ | |
| 2 | f1ofo 6855 | . . 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 7193 | . . . 4 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥) | |
| 5 | fvresi 7193 | . . . 4 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦) | |
| 6 | 4, 5 | oveqan12d 7450 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)) | 
| 7 | 6 | rgen2 3199 | . 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦) | 
| 8 | elunop 31891 | . 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 1540 ∈ wcel 2108 ∀wral 3061 I cid 5577 ↾ cres 5687 –onto→wfo 6559 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℋchba 30938 ·ih csp 30941 UniOpcuo 30968 | 
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hilex 31018 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-unop 31862 | 
| This theorem is referenced by: idlnop 32011 | 
| Copyright terms: Public domain | W3C validator |