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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 6873 | . . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ | |
2 | f1ofo 6842 | . . 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 7179 | . . . 4 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥) | |
5 | fvresi 7179 | . . . 4 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦) | |
6 | 4, 5 | oveqan12d 7435 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)) |
7 | 6 | rgen2 3188 | . 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦) |
8 | elunop 31802 | . 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 709 | 1 ⊢ ( I ↾ ℋ) ∈ UniOp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ∀wral 3051 I cid 5571 ↾ cres 5676 –onto→wfo 6544 –1-1-onto→wf1o 6545 ‘cfv 6546 (class class class)co 7416 ℋchba 30849 ·ih csp 30852 UniOpcuo 30879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-hilex 30929 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-unop 31773 |
This theorem is referenced by: idlnop 31922 |
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