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Theorem hmopco 30965

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 30816	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2		hmopf 30816	. . . 4 ⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3		fco 6692	. . . 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 1132	. 2 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
6		fvco3 6940	. . . . . . . . . 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 7373	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))))
9	8	ad2ant2l 744	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))))
10		simpll 765	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp)
11		simprl 769	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
12	2	ffvelcdmda 7035	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈‘𝑦) ∈ ℋ)
13	12	ad2ant2l 744	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈‘𝑦) ∈ ℋ)
14		hmop 30864	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)))
15	10, 11, 13, 14	syl3anc 1371	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)))
16		simplr 767	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
17	1	ffvelcdmda 7035	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
18	17	ad2ant2r 745	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇‘𝑥) ∈ ℋ)
19		simprr 771	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20		hmop 30864	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
21	16, 18, 19, 20	syl3anc 1371	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·_ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
22	9, 15, 21	3eqtrd 2780	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
23		fvco3 6940	. . . . . . . . 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 7372	. . . . . . 7 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
26	25	ad2ant2r 745	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·_ih 𝑦))
27	22, 26	eqtr4d 2779	. . . . 5 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
28	27	3adantl3 1168	. . . 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 6841	. . . . . . 7 ⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → ((𝑇 ∘ 𝑈)‘𝑥) = ((𝑈 ∘ 𝑇)‘𝑥))
30	29	oveq1d 7372	. . . . . 6 ⊢ ((𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
31	30	3ad2ant3 1135	. . . . 5 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (((𝑇 ∘ 𝑈)‘𝑥) ·_ih 𝑦) = (((𝑈 ∘ 𝑇)‘𝑥) ·_ih 𝑦))
32	31	adantr 481	. . . 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 2779	. . 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 3197	. 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 30815	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∘ ccom 5637 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℋchba 29861 ·_ih csp 29864 HrmOpcho 29892

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-hilex 29941

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-map 8767 df-hmop 30786

This theorem is referenced by: leopsq 31071 opsqrlem4 31085 opsqrlem6 31087

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