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Theorem idunop 31998
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 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)
31, 2ax-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)
64, 5oveqan12d 7451 . . 3 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦))
76rgen2 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)))
93, 7, 8mpbir2an 711 1 ( I ↾ ℋ) ∈ UniOp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  wral 3060   I cid 5576  cres 5686  ontowfo 6558  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  chba 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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