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Theorem hmopco 28728
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 ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈) ∈ HrmOp)

Proof of Theorem hmopco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmopf 28579 . . . 4 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2 hmopf 28579 . . . 4 (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3 fco 6015 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇𝑈): ℋ⟶ ℋ)
41, 2, 3syl2an 494 . . 3 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇𝑈): ℋ⟶ ℋ)
543adant3 1079 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈): ℋ⟶ ℋ)
6 fvco3 6232 . . . . . . . . . 10 ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
72, 6sylan 488 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
87oveq2d 6620 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
98ad2ant2l 781 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
10 simpll 789 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp)
11 simprl 793 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
122ffvelrnda 6315 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈𝑦) ∈ ℋ)
1312ad2ant2l 781 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈𝑦) ∈ ℋ)
14 hmop 28627 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑈𝑦))) = ((𝑇𝑥) ·ih (𝑈𝑦)))
1510, 11, 13, 14syl3anc 1323 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇‘(𝑈𝑦))) = ((𝑇𝑥) ·ih (𝑈𝑦)))
16 simplr 791 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
171ffvelrnda 6315 . . . . . . . . 9 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
1817ad2ant2r 782 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
19 simprr 795 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20 hmop 28627 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ (𝑇𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih (𝑈𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2116, 18, 19, 20syl3anc 1323 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih (𝑈𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
229, 15, 213eqtrd 2659 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
23 fvco3 6232 . . . . . . . . 9 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
241, 23sylan 488 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
2524oveq1d 6619 . . . . . . 7 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2625ad2ant2r 782 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2722, 26eqtr4d 2658 . . . . 5 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
28273adantl3 1217 . . . 4 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
29 fveq1 6147 . . . . . . 7 ((𝑇𝑈) = (𝑈𝑇) → ((𝑇𝑈)‘𝑥) = ((𝑈𝑇)‘𝑥))
3029oveq1d 6619 . . . . . 6 ((𝑇𝑈) = (𝑈𝑇) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
31303ad2ant3 1082 . . . . 5 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
3231adantr 481 . . . 4 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
3328, 32eqtr4d 2658 . . 3 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
3433ralrimivva 2965 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
35 elhmop 28578 . 2 ((𝑇𝑈) ∈ HrmOp ↔ ((𝑇𝑈): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦)))
365, 34, 35sylanbrc 697 1 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈) ∈ HrmOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  chil 27622   ·ih csp 27625  HrmOpcho 27653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-hilex 27702
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-hmop 28549
This theorem is referenced by:  leopsq  28834  opsqrlem4  28848  opsqrlem6  28850
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