Theorem idunop 32007

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.

Step	Hyp	Ref	Expression
1		f1oi 6887	. . 3 ⊢ ( I ↾ ℋ): ℋ–1-1-onto→ ℋ
2		f1ofo 6856	. . 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 7193	. . . 4 ⊢ (𝑥 ∈ ℋ → (( I ↾ ℋ)‘𝑥) = 𝑥)
5		fvresi 7193	. . . 4 ⊢ (𝑦 ∈ ℋ → (( I ↾ ℋ)‘𝑦) = 𝑦)
6	4, 5	oveqan12d 7450	. . 3 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦))
7	6	rgen2 3197	. 2 ⊢ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((( I ↾ ℋ)‘𝑥) ·ih (( I ↾ ℋ)‘𝑦)) = (𝑥 ·ih 𝑦)
8		elunop 31901	. 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 711	1 ⊢ ( I ↾ ℋ) ∈ UniOp

Syntax hints:   = wceq 1537   ∈ wcel 2106  ∀wral 3059   I cid 5582   ↾ cres 5691  –onto→wfo 6561  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431   ℋchba 30948   ·_ih csp 30951  UniOpcuo 30978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-hilex 31028

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-unop 31872

This theorem is referenced by:  idlnop  32021