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Theorem idunop 28698
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 6133 . . 3 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2 f1ofo 6103 . . 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 6396 . . . 4 (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
5 fvresi 6396 . . . 4 (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
64, 5oveqan12d 6626 . . 3 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦))
76rgen2a 2971 . 2 𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)
8 elunop 28592 . 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 954 1 ( I ↾ ℋ) ∈ UniOp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wral 2907   I cid 4986  cres 5078  ontowfo 5847  1-1-ontowf1o 5848  cfv 5849  (class class class)co 6607  chil 27637   ·ih csp 27640  UniOpcuo 27667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-hilex 27717
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-unop 28563
This theorem is referenced by:  idlnop  28712
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