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Theorem idunop 29551
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 6486 . . 3 ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2 f1ofo 6456 . . 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 6764 . . . 4 (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
5 fvresi 6764 . . . 4 (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
64, 5oveqan12d 7001 . . 3 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦))
76rgen2a 3178 . 2 𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)
8 elunop 29445 . 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 699 1 ( I ↾ ℋ) ∈ UniOp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1508  wcel 2051  wral 3090   I cid 5315  cres 5413  ontowfo 6191  1-1-ontowf1o 6192  cfv 6193  (class class class)co 6982  chba 28490   ·ih csp 28493  UniOpcuo 28520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-rep 5053  ax-sep 5064  ax-nul 5071  ax-pr 5190  ax-hilex 28570
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-iun 4799  df-br 4935  df-opab 4997  df-mpt 5014  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-ov 6985  df-unop 29416
This theorem is referenced by:  idlnop  29565
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