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Theorem hmopco 28266
 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 28117 . . . 4 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2 hmopf 28117 . . . 4 (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3 fco 5971 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇𝑈): ℋ⟶ ℋ)
41, 2, 3syl2an 493 . . 3 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇𝑈): ℋ⟶ ℋ)
543adant3 1074 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈): ℋ⟶ ℋ)
6 fvco3 6185 . . . . . . . . . 10 ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
72, 6sylan 487 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
87oveq2d 6565 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
98ad2ant2l 778 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
10 simpll 786 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp)
11 simprl 790 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
122ffvelrnda 6267 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈𝑦) ∈ ℋ)
1312ad2ant2l 778 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈𝑦) ∈ ℋ)
14 hmop 28165 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑈𝑦))) = ((𝑇𝑥) ·ih (𝑈𝑦)))
1510, 11, 13, 14syl3anc 1318 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇‘(𝑈𝑦))) = ((𝑇𝑥) ·ih (𝑈𝑦)))
16 simplr 788 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
171ffvelrnda 6267 . . . . . . . . 9 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
1817ad2ant2r 779 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
19 simprr 792 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20 hmop 28165 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ (𝑇𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih (𝑈𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2116, 18, 19, 20syl3anc 1318 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih (𝑈𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
229, 15, 213eqtrd 2648 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
23 fvco3 6185 . . . . . . . . 9 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
241, 23sylan 487 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
2524oveq1d 6564 . . . . . . 7 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2625ad2ant2r 779 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2722, 26eqtr4d 2647 . . . . 5 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
28273adantl3 1212 . . . 4 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
29 fveq1 6102 . . . . . . 7 ((𝑇𝑈) = (𝑈𝑇) → ((𝑇𝑈)‘𝑥) = ((𝑈𝑇)‘𝑥))
3029oveq1d 6564 . . . . . 6 ((𝑇𝑈) = (𝑈𝑇) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
31303ad2ant3 1077 . . . . 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 2647 . . 3 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
3433ralrimivva 2954 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
35 elhmop 28116 . 2 ((𝑇𝑈) ∈ HrmOp ↔ ((𝑇𝑈): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦)))
365, 34, 35sylanbrc 695 1 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈) ∈ HrmOp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ℋchil 27160   ·ih csp 27163  HrmOpcho 27191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-hmop 28087 This theorem is referenced by:  leopsq  28372  opsqrlem4  28386  opsqrlem6  28388
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