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Theorem hmopco 30494
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 30345 . . . 4 (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)
2 hmopf 30345 . . . 4 (𝑈 ∈ HrmOp → 𝑈: ℋ⟶ ℋ)
3 fco 6661 . . . 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 6906 . . . . . . . . . 10 ((𝑈: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
72, 6sylan 580 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → ((𝑇𝑈)‘𝑦) = (𝑇‘(𝑈𝑦)))
87oveq2d 7331 . . . . . . . 8 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
98ad2ant2l 743 . . . . . . 7 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (𝑥 ·ih (𝑇‘(𝑈𝑦))))
10 simpll 764 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑇 ∈ HrmOp)
11 simprl 768 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑥 ∈ ℋ)
122ffvelcdmda 7000 . . . . . . . . 9 ((𝑈 ∈ HrmOp ∧ 𝑦 ∈ ℋ) → (𝑈𝑦) ∈ ℋ)
1312ad2ant2l 743 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑈𝑦) ∈ ℋ)
14 hmop 30393 . . . . . . . 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 766 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑈 ∈ HrmOp)
171ffvelcdmda 7000 . . . . . . . . 9 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
1817ad2ant2r 744 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑇𝑥) ∈ ℋ)
19 simprr 770 . . . . . . . 8 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → 𝑦 ∈ ℋ)
20 hmop 30393 . . . . . . . 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 2781 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
23 fvco3 6906 . . . . . . . . 9 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
241, 23sylan 580 . . . . . . . 8 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → ((𝑈𝑇)‘𝑥) = (𝑈‘(𝑇𝑥)))
2524oveq1d 7330 . . . . . . 7 ((𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2625ad2ant2r 744 . . . . . 6 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑈𝑇)‘𝑥) ·ih 𝑦) = ((𝑈‘(𝑇𝑥)) ·ih 𝑦))
2722, 26eqtr4d 2780 . . . . 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 6810 . . . . . . 7 ((𝑇𝑈) = (𝑈𝑇) → ((𝑇𝑈)‘𝑥) = ((𝑈𝑇)‘𝑥))
3029oveq1d 7330 . . . . . 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 481 . . . 4 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇𝑈)‘𝑥) ·ih 𝑦) = (((𝑈𝑇)‘𝑥) ·ih 𝑦))
3328, 32eqtr4d 2780 . . 3 (((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
3433ralrimivva 3194 . 2 ((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp ∧ (𝑇𝑈) = (𝑈𝑇)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih ((𝑇𝑈)‘𝑦)) = (((𝑇𝑈)‘𝑥) ·ih 𝑦))
35 elhmop 30344 . 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 396  w3a 1086   = wceq 1540  wcel 2105  wral 3062  ccom 5611  wf 6461  cfv 6465  (class class class)co 7315  chba 29390   ·ih csp 29393  HrmOpcho 29421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628  ax-hilex 29470
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fv 6473  df-ov 7318  df-oprab 7319  df-mpo 7320  df-map 8665  df-hmop 30315
This theorem is referenced by:  leopsq  30600  opsqrlem4  30614  opsqrlem6  30616
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