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Theorem hmopco 32353

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 32204	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2		hmopf 32204	. . . 4 ⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3		fco 6732	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
4	1, 2, 3	syl2an 607	. . 3 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
5	4	3adant3 1150	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
6		fvco3 6983	. . . . . . . . . 10 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦)))
7	2, 6	sylan 591	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦)))
8	7	oveq2d 7428	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))))
9	8	ad2ant2l 758	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))))
10		simpll 778	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp)
11		simprl 782	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
12	2	ffvelcdmda 7081	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈‘𝑦) ∈ ℋ)
13	12	ad2ant2l 758	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈‘𝑦) ∈ ℋ)
14		hmop 32252	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)))
15	10, 11, 13, 14	syl3anc 1398	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)))
16		simplr 780	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
17	1	ffvelcdmda 7081	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
18	17	ad2ant2r 759	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ)
19		simprr 784	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20		hmop 32252	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
21	16, 18, 19, 20	syl3anc 1398	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
22	9, 15, 21	3eqtrd 2802	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
23		fvco3 6983	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥)))
24	1, 23	sylan 591	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥)))
25	24	oveq1d 7427	. . . . . . 7 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
26	25	ad2ant2r 759	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
27	22, 26	eqtr4d 2801	. . . . 5 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
28	27	3adantl3 1187	. . . 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 6882	. . . . . . 7 ⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → ((𝑇 ∘ 𝑈)‘𝑥) = ((𝑈 ∘ 𝑇)‘𝑥))
30	29	oveq1d 7427	. . . . . 6 ⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
31	30	3ad2ant3 1153	. . . . 5 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
32	31	adantr 485	. . . 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 2801	. . 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 3208	. 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 32203	. 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 594	1 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈) ∈ HrmOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∘ ccom 5667 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 ℋchba 31249 ·_ih csp 31252 HrmOpcho 31280

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-hilex 31329

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-map 8827 df-hmop 32174

This theorem is referenced by: leopsq 32459 opsqrlem4 32473 opsqrlem6 32475

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