elhmop - Hilbert Space Explorer

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Theorem elhmop 29304

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 6445	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
2	1	oveq2d 6938	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑥 ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
3		fveq1 6445	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
4	3	oveq1d 6937	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·_ih 𝑦) = ((𝑇‘𝑥) ·_ih 𝑦))
5	2, 4	eqeq12d 2793	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
6	5	2ralbidv 3171	. . 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 29275	. . 3 ⊢ HrmOp = {𝑡 ∈ ( ℋ ↑_𝑚 ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦)}
8	6, 7	elrab2 3576	. 2 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇 ∈ ( ℋ ↑_𝑚 ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
9		ax-hilex 28428	. . . 4 ⊢ ℋ ∈ V
10	9, 9	elmap 8169	. . 3 ⊢ (𝑇 ∈ ( ℋ ↑_𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
11	10	anbi1i 617	. 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 267	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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↑_𝑚 cmap 8140 ℋchba 28348 ·_ih csp 28351 HrmOpcho 28379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-hilex 28428

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-map 8142 df-hmop 29275

This theorem is referenced by: hmopf 29305 hmop 29353 hmopadj2 29372 idhmop 29413 0hmop 29414 lnophmi 29449 hmops 29451 hmopm 29452 hmopco 29454 pjhmopi 29577

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