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Theorem hmopco 29813
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 29664 . . . 4 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2 hmopf 29664 . . . 4 (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3 fco 6521 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑈: ℋ⟶ ℋ) → (𝑇𝑈): ℋ⟶ ℋ)
41, 2, 3syl2an 598 . . 3 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇𝑈): ℋ⟶ ℋ)
543adant3 1129 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈): ℋ⟶ ℋ)
6 fvco3 6751 . . . . . . . . . 10 ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
72, 6sylan 583 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
87oveq2d 7165 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
98ad2ant2l 745 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
10 simpll 766 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp)
11 simprl 770 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
122ffvelrnda 6842 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈𝑦) ∈ ℋ)
1312ad2ant2l 745 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈𝑦) ∈ ℋ)
14 hmop 29712 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ∧ (𝑈𝑦) ∈ ℋ) → (𝑥 ·ih (𝑇‘(𝑈𝑦))) = ((𝑇𝑥) ·ih (𝑈𝑦)))
1510, 11, 13, 14syl3anc 1368 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih (𝑇‘(𝑈𝑦))) = ((𝑇𝑥) ·ih (𝑈𝑦)))
16 simplr 768 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
171ffvelrnda 6842 . . . . . . . . 9 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
1817ad2ant2r 746 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
19 simprr 772 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20 hmop 29712 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ (𝑇𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑥) ·ih (𝑈𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2116, 18, 19, 20syl3anc 1368 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇𝑥) ·ih (𝑈𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
229, 15, 213eqtrd 2863 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
23 fvco3 6751 . . . . . . . . 9 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
241, 23sylan 583 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
2524oveq1d 7164 . . . . . . 7 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2625ad2ant2r 746 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2722, 26eqtr4d 2862 . . . . 5 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
28273adantl3 1165 . . . 4 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
29 fveq1 6660 . . . . . . 7 ((𝑇𝑈) = (𝑈𝑇) → ((𝑇𝑈)‘𝑥) = ((𝑈𝑇)‘𝑥))
3029oveq1d 7164 . . . . . 6 ((𝑇𝑈) = (𝑈𝑇) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
31303ad2ant3 1132 . . . . 5 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
3231adantr 484 . . . 4 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
3328, 32eqtr4d 2862 . . 3 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
3433ralrimivva 3186 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
35 elhmop 29663 . 2 ((𝑇𝑈) ∈ HrmOp ↔ ((𝑇𝑈): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦)))
365, 34, 35sylanbrc 586 1 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → (𝑇𝑈) ∈ HrmOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  ccom 5546  wf 6339  cfv 6343  (class class class)co 7149  chba 28709   ·ih csp 28712  HrmOpcho 28740
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-hilex 28789
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-hmop 29634
This theorem is referenced by:  leopsq  29919  opsqrlem4  29933  opsqrlem6  29935
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