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Theorem hmopco 32042

Description: The composition of two commuting Hermitian operators is Hermitian. (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hmopco	⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈) ∈ HrmOp)

Proof of Theorem hmopco Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmopf 31893	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2		hmopf 31893	. . . 4 ⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3		fco 6760	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
4	1, 2, 3	syl2an 596	. . 3 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
5	4	3adant3 1133	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
6		fvco3 7008	. . . . . . . . . 10 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦)))
7	2, 6	sylan 580	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦)))
8	7	oveq2d 7447	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))))
9	8	ad2ant2l 746	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))))
10		simpll 767	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp)
11		simprl 771	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
12	2	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈‘𝑦) ∈ ℋ)
13	12	ad2ant2l 746	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈‘𝑦) ∈ ℋ)
14		hmop 31941	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)))
15	10, 11, 13, 14	syl3anc 1373	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)))
16		simplr 769	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
17	1	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
18	17	ad2ant2r 747	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ)
19		simprr 773	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20		hmop 31941	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
21	16, 18, 19, 20	syl3anc 1373	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
22	9, 15, 21	3eqtrd 2781	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
23		fvco3 7008	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥)))
24	1, 23	sylan 580	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥)))
25	24	oveq1d 7446	. . . . . . 7 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
26	25	ad2ant2r 747	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
27	22, 26	eqtr4d 2780	. . . . 5 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
28	27	3adantl3 1169	. . . 4 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
29		fveq1 6905	. . . . . . 7 ⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → ((𝑇 ∘ 𝑈)‘𝑥) = ((𝑈 ∘ 𝑇)‘𝑥))
30	29	oveq1d 7446	. . . . . 6 ⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
31	30	3ad2ant3 1136	. . . . 5 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
32	31	adantr 480	. . . 4 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
33	28, 32	eqtr4d 2780	. . 3 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦))
34	33	ralrimivva 3202	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦))
35		elhmop 31892	. 2 ⊢ ((𝑇 ∘ 𝑈) ∈ HrmOp ↔ ((𝑇 ∘ 𝑈): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦)))
36	5, 34, 35	sylanbrc 583	1 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈) ∈ HrmOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℋchba 30938 ·_ih csp 30941 HrmOpcho 30969

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-hilex 31018

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-hmop 31863

This theorem is referenced by: leopsq 32148 opsqrlem4 32162 opsqrlem6 32164

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