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Theorem hmopco 32051
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 31902 . . . 4 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2 hmopf 31902 . . . 4 (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3 fco 6760 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇𝑈): ℋ⟶ ℋ)
41, 2, 3syl2an 596 . . 3 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇𝑈): ℋ⟶ ℋ)
543adant3 1131 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈): ℋ⟶ ℋ)
6 fvco3 7007 . . . . . . . . . 10 ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
72, 6sylan 580 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
87oveq2d 7446 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
98ad2ant2l 746 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
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)
122ffvelcdmda 7103 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈𝑦) ∈ ℋ)
1312ad2ant2l 746 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈𝑦) ∈ ℋ)
14 hmop 31950 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑈𝑦))) = ((𝑇𝑥) ·ih (𝑈𝑦)))
1510, 11, 13, 14syl3anc 1370 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇‘(𝑈𝑦))) = ((𝑇𝑥) ·ih (𝑈𝑦)))
16 simplr 769 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
171ffvelcdmda 7103 . . . . . . . . 9 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
1817ad2ant2r 747 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
19 simprr 773 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20 hmop 31950 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ (𝑇𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih (𝑈𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2116, 18, 19, 20syl3anc 1370 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih (𝑈𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
229, 15, 213eqtrd 2778 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
23 fvco3 7007 . . . . . . . . 9 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
241, 23sylan 580 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
2524oveq1d 7445 . . . . . . 7 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2625ad2ant2r 747 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2722, 26eqtr4d 2777 . . . . 5 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
28273adantl3 1167 . . . 4 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
29 fveq1 6905 . . . . . . 7 ((𝑇𝑈) = (𝑈𝑇) → ((𝑇𝑈)‘𝑥) = ((𝑈𝑇)‘𝑥))
3029oveq1d 7445 . . . . . 6 ((𝑇𝑈) = (𝑈𝑇) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
31303ad2ant3 1134 . . . . 5 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
3231adantr 480 . . . 4 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
3328, 32eqtr4d 2777 . . 3 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
3433ralrimivva 3199 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
35 elhmop 31901 . 2 ((𝑇𝑈) ∈ HrmOp ↔ ((𝑇𝑈): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦)))
365, 34, 35sylanbrc 583 1 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈) ∈ HrmOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  ccom 5692  wf 6558  cfv 6562  (class class class)co 7430  chba 30947   ·ih csp 30950  HrmOpcho 30978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-hilex 31027
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-hmop 31872
This theorem is referenced by:  leopsq  32157  opsqrlem4  32171  opsqrlem6  32173
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