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Theorem elhmop 31948

Description: Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
elhmop	⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))

Distinct variable group: 𝑥,𝑦,𝑇

Proof of Theorem elhmop Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6833	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
2	1	oveq2d 7374	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑥 ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
3		fveq1 6833	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
4	3	oveq1d 7373	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·_ih 𝑦) = ((𝑇‘𝑥) ·_ih 𝑦))
5	2, 4	eqeq12d 2752	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
6	5	2ralbidv 3200	. . 3 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
7		df-hmop 31919	. . 3 ⊢ HrmOp = {𝑡 ∈ ( ℋ ↑_m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)}
8	6, 7	elrab2 3649	. 2 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
9		ax-hilex 31074	. . . 4 ⊢ ℋ ∈ V
10	9, 9	elmap 8809	. . 3 ⊢ (𝑇 ∈ ( ℋ ↑_m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
11	10	anbi1i 624	. 2 ⊢ ((𝑇 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
12	8, 11	bitri 275	1 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ↑_m cmap 8763 ℋchba 30994 ·_ih csp 30997 HrmOpcho 31025

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-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 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-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-map 8765 df-hmop 31919

This theorem is referenced by: hmopf 31949 hmop 31997 hmopadj2 32016 idhmop 32057 0hmop 32058 lnophmi 32093 hmops 32095 hmopm 32096 hmopco 32098 pjhmopi 32221

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