Theorem elunop 31947

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 3461	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ V)
2		fof 6746	. . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3		ax-hilex 31074	. . . 4 ⊢ ℋ ∈ V
4		fex 7172	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑇 ∈ V)
5	2, 3, 4	sylancl 586	. . 3 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇 ∈ V)
6	5	adantr 480	. 2 ⊢ ((𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)) → 𝑇 ∈ V)
7		foeq1 6742	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑡: ℋ–onto→ ℋ ↔ 𝑇: ℋ–onto→ ℋ))
8		fveq1 6833	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
9		fveq1 6833	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
10	8, 9	oveq12d 7376	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·ih (𝑇‘𝑦)))
11	10	eqeq1d 2738	. . . . 5 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦) ↔ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))
12	11	2ralbidv 3200	. . . 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 31918	. . 3 ⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))}
15	13, 14	elab2g 3635	. 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 378	1 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358   ℋchba 30994   ·_ih csp 30997  UniOpcuo 31024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-hilex 31074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-unop 31918

This theorem is referenced by:  unop  31990  unopf1o  31991  cnvunop  31993  counop  31996  idunop  32053  lnopunii  32087  elunop2  32088