Theorem hmopco 32110

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 31961	. . . 4 ⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2		hmopf 31961	. . . 4 ⊢ (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3		fco 6694	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇 ∘ 𝑈): ℋ⟶ ℋ)
4	1, 2, 3	syl2an 597	. . 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 6941	. . . . . . . . . 10 ⊢ ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦)))
7	2, 6	sylan 581	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇 ∘ 𝑈)‘𝑦) = (𝑇‘(𝑈‘𝑦)))
8	7	oveq2d 7384	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈‘𝑦))))
9	8	ad2ant2l 747	. . . . . . 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 7038	. . . . . . . . 9 ⊢ ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈‘𝑦) ∈ ℋ)
13	12	ad2ant2l 747	. . . . . . . 8 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈‘𝑦) ∈ ℋ)
14		hmop 32009	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈‘𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑈‘𝑦))) = ((𝑇‘𝑥) ·ih (𝑈‘𝑦)))
15	10, 11, 13, 14	syl3anc 1374	. . . . . . 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 7038	. . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
18	17	ad2ant2r 748	. . . . . . . 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 32009	. . . . . . . 8 ⊢ ((𝑈 ∈ HrmOp ∧ (𝑇‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦))
21	16, 18, 19, 20	syl3anc 1374	. . . . . . 7 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·ih (𝑈‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦))
22	9, 15, 21	3eqtrd 2776	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇 ∘ 𝑈)‘𝑦)) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦))
23		fvco3 6941	. . . . . . . . 9 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥)))
24	1, 23	sylan 581	. . . . . . . 8 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈 ∘ 𝑇)‘𝑥) = (𝑈‘(𝑇‘𝑥)))
25	24	oveq1d 7383	. . . . . . 7 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦))
26	25	ad2ant2r 748	. . . . . 6 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇‘𝑥)) ·ih 𝑦))
27	22, 26	eqtr4d 2775	. . . . 5 ⊢ (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇 ∘ 𝑈)‘𝑦)) = (((𝑈 ∘ 𝑇)‘𝑥) ·ih 𝑦))
28	27	3adantl3 1170	. . . 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 7383	. . . . . 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 2775	. . 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 3181	. 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 31960	. 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 584	1 ⊢ ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇 ∘ 𝑈) = (𝑈 ∘ 𝑇)) → (𝑇 ∘ 𝑈) ∈ HrmOp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∘ ccom 5636  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ℋchba 31006   ·_ih csp 31009  HrmOpcho 31037

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-hilex 31086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-hmop 31931

This theorem is referenced by:  leopsq  32216  opsqrlem4  32230  opsqrlem6  32232