Theorem elunop 31156

Description: Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
elunop	⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))

Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem elunop

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3493	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ V)
2		fof 6806	. . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3		ax-hilex 30283	. . . 4 ⊢ ℋ ∈ V
4		fex 7228	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑇 ∈ V)
5	2, 3, 4	sylancl 587	. . 3 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇 ∈ V)
6	5	adantr 482	. 2 ⊢ ((𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) → 𝑇 ∈ V)
7		foeq1 6802	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑡: ℋ–onto→ ℋ ↔ 𝑇: ℋ–onto→ ℋ))
8		fveq1 6891	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
9		fveq1 6891	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
10	8, 9	oveq12d 7427	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·ih (𝑇‘𝑦)))
11	10	eqeq1d 2735	. . . . 5 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))
12	11	2ralbidv 3219	. . . 4 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))
13	7, 12	anbi12d 632	. . 3 ⊢ (𝑡 = 𝑇 → ((𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦)) ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))))
14		df-unop 31127	. . 3 ⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))}
15	13, 14	elab2g 3671	. 2 ⊢ (𝑇 ∈ V → (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))))
16	1, 6, 15	pm5.21nii 380	1 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  ⟶wf 6540  –onto→wfo 6542  ‘cfv 6544  (class class class)co 7409   ℋchba 30203   ·_ih csp 30206  UniOpcuo 30233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-hilex 30283

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-unop 31127

This theorem is referenced by:  unop  31199  unopf1o  31200  cnvunop  31202  counop  31205  idunop  31262  lnopunii  31296  elunop2  31297