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Theorem idunop 29764
Description: The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
idunop ( I ↾ ℋ) ∈ UniOp

Proof of Theorem idunop
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 6631 . . 3 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2 f1ofo 6601 . . 3 (( I ↾ ℋ): ℋ–1-1-onto→ ℋ → ( I ↾ ℋ): ℋ–onto→ ℋ)
31, 2ax-mp 5 . 2 ( I ↾ ℋ): ℋ–onto→ ℋ
4 fvresi 6916 . . . 4 (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
5 fvresi 6916 . . . 4 (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
64, 5oveqan12d 7158 . . 3 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦))
76rgen2 3171 . 2 𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)
8 elunop 29658 . 2 (( I ↾ ℋ) ∈ UniOp ↔ (( I ↾ ℋ): ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)))
93, 7, 8mpbir2an 710 1 ( I ↾ ℋ) ∈ UniOp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2112  wral 3109   I cid 5427  cres 5525  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  chba 28705   ·ih csp 28708  UniOpcuo 28735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-hilex 28785
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-unop 29629
This theorem is referenced by:  idlnop  29778
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