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Theorem hmopco 32172

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 32023	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2		hmopf 32023	. . . 4 ⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3		fco 6712	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
4	1, 2, 3	syl2an 605	. . 3 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
5	4	3adant3 1144	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
6		fvco3 6963	. . . . . . . . . 10 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦)))
7	2, 6	sylan 589	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦)))
8	7	oveq2d 7408	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))))
9	8	ad2ant2l 756	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))))
10		simpll 776	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp)
11		simprl 780	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
12	2	ffvelcdmda 7061	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈‘𝑦) ∈ ℋ)
13	12	ad2ant2l 756	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈‘𝑦) ∈ ℋ)
14		hmop 32071	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)))
15	10, 11, 13, 14	syl3anc 1389	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)))
16		simplr 778	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
17	1	ffvelcdmda 7061	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
18	17	ad2ant2r 757	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ)
19		simprr 782	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20		hmop 32071	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
21	16, 18, 19, 20	syl3anc 1389	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
22	9, 15, 21	3eqtrd 2800	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
23		fvco3 6963	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥)))
24	1, 23	sylan 589	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥)))
25	24	oveq1d 7407	. . . . . . 7 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
26	25	ad2ant2r 757	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
27	22, 26	eqtr4d 2799	. . . . 5 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
28	27	3adantl3 1181	. . . 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 6862	. . . . . . 7 ⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → ((𝑇 ∘ 𝑈)‘𝑥) = ((𝑈 ∘ 𝑇)‘𝑥))
30	29	oveq1d 7407	. . . . . 6 ⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
31	30	3ad2ant3 1147	. . . . 5 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
32	31	adantr 484	. . . 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 2799	. . 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 3204	. 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 32022	. 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 592	1 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈) ∈ HrmOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∘ ccom 5649 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 ℋchba 31068 ·_ih csp 31071 HrmOpcho 31099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-hilex 31148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-map 8805 df-hmop 31993

This theorem is referenced by: leopsq 32278 opsqrlem4 32292 opsqrlem6 32294

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