elunop - Hilbert Space Explorer

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Theorem elunop 31805

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 3482	. 2 ⊢ (𝑇 ∈ UniOp → 𝑇 ∈ V)
2		fof 6815	. . . 4 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇: ℋ⟶ ℋ)
3		ax-hilex 30932	. . . 4 ⊢ ℋ ∈ V
4		fex 7243	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ℋ ∈ V) → 𝑇 ∈ V)
5	2, 3, 4	sylancl 584	. . 3 ⊢ (𝑇: ℋ–onto→ ℋ → 𝑇 ∈ V)
6	5	adantr 479	. 2 ⊢ ((𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)) → 𝑇 ∈ V)
7		foeq1 6811	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑡: ℋ–onto→ ℋ ↔ 𝑇: ℋ–onto→ ℋ))
8		fveq1 6900	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
9		fveq1 6900	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
10	8, 9	oveq12d 7442	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·_ih (𝑡‘𝑦)) = ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)))
11	10	eqeq1d 2728	. . . . 5 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥) ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih 𝑦) ↔ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))
12	11	2ralbidv 3209	. . . 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 630	. . 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 31776	. . 3 ⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih 𝑦))}
15	13, 14	elab2g 3668	. 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 377	1 ⊢ (𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ⟶wf 6550 –onto→wfo 6552 ‘cfv 6554 (class class class)co 7424 ℋchba 30852 ·_ih csp 30855 UniOpcuo 30882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-hilex 30932

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-unop 31776

This theorem is referenced by: unop 31848 unopf1o 31849 cnvunop 31851 counop 31854 idunop 31911 lnopunii 31945 elunop2 31946

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