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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 6174 | . . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ | |
2 | f1ofo 6144 | . . 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 6439 | . . . 4 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥) | |
5 | fvresi 6439 | . . . 4 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦) | |
6 | 4, 5 | oveqan12d 6669 | . . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)) |
7 | 6 | rgen2a 2977 | . 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦) |
8 | elunop 28731 | . 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 955 | 1 ⊢ ( I ↾ ℋ) ∈ UniOp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ∀wral 2912 I cid 5023 ↾ cres 5116 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ℋchil 27776 ·ih csp 27779 UniOpcuo 27806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-unop 28702 |
This theorem is referenced by: idlnop 28851 |
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