elhmop - Hilbert Space Explorer

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Theorem elhmop 32475

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 6884	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
2	1	oveq2d 7436	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑥 ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
3		fveq1 6884	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
4	3	oveq1d 7435	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥) ·_ih 𝑦) = ((𝑇‘𝑥) ·_ih 𝑦))
5	2, 4	eqeq12d 2777	. . . 4 ⊢ (𝑡 = 𝑇 → ((𝑥 ·_ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
6	5	2ralbidv 3227	. . 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 32446	. . 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 31601	. . . 4 ⊢ ℋ ∈ V
10	9, 9	elmap 8899	. . 3 ⊢ (𝑇 ∈ ( ℋ ↑_m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
11	10	anbi1i 636	. 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 278	1 ⊢ (𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ↑_m cmap 8847 ℋchba 31521 ·_ih csp 31524 HrmOpcho 31552

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-hilex 31601

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8849 df-hmop 32446

This theorem is used by: hmopf 32476 hmop 32524 hmopadj2 32543 idhmop 32584 0hmop 32585 lnophmi 32620 hmops 32622 hmopm 32623 hmopco 32625 pjhmopi 32748

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