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Theorem lnopmi 29787
Description: The scalar product of a linear operator is a linear operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
lnopm.1 𝑇 ∈ LinOp
Assertion
Ref Expression
lnopmi (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)

Proof of Theorem lnopmi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnopm.1 . . . 4 𝑇 ∈ LinOp
21lnopfi 29756 . . 3 𝑇: ℋ⟶ ℋ
3 homulcl 29546 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
42, 3mpan2 690 . 2 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5 hvmulcl 28800 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
6 hvaddcl 28799 . . . . . . . 8 (((𝑥 · 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
75, 6sylan 583 . . . . . . 7 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
8 homval 29528 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
92, 8mp3an2 1446 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
107, 9sylan2 595 . . . . . 6 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
11 id 22 . . . . . . . . 9 (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
122ffvelrni 6831 . . . . . . . . . 10 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
13 hvmulcl 28800 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
1412, 13sylan2 595 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
152ffvelrni 6831 . . . . . . . . 9 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
16 ax-hvdistr1 28795 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 · (𝑇𝑦)) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
1711, 14, 15, 16syl3an 1157 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
18173expb 1117 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
191lnopli 29755 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
20193expa 1115 . . . . . . . . 9 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
2120oveq2d 7155 . . . . . . . 8 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
2221adantl 485 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
23 homval 29528 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
242, 23mp3an2 1446 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2524adantrl 715 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2625oveq2d 7155 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 · (𝐴 · (𝑇𝑦))))
27 hvmulcom 28830 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
2812, 27syl3an3 1162 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
29283expb 1117 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
3026, 29eqtr4d 2839 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 · (𝑥 · (𝑇𝑦))))
31 homval 29528 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
322, 31mp3an2 1446 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
3330, 32oveqan12d 7158 . . . . . . . 8 (((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ (𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ)) → ((𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) + ((𝐴 ·op 𝑇)‘𝑧)) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
3433anandis 677 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) + ((𝐴 ·op 𝑇)‘𝑧)) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
3518, 22, 343eqtr4rd 2847 . . . . . 6 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) + ((𝐴 ·op 𝑇)‘𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
3610, 35eqtr4d 2839 . . . . 5 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) + ((𝐴 ·op 𝑇)‘𝑧)))
3736exp32 424 . . . 4 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑧 ∈ ℋ → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) + ((𝐴 ·op 𝑇)‘𝑧)))))
3837ralrimdv 3156 . . 3 (𝐴 ∈ ℂ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ∀𝑧 ∈ ℋ ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) + ((𝐴 ·op 𝑇)‘𝑧))))
3938ralrimivv 3158 . 2 (𝐴 ∈ ℂ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) + ((𝐴 ·op 𝑇)‘𝑧)))
40 ellnop 29645 . 2 ((𝐴 ·op 𝑇) ∈ LinOp ↔ ((𝐴 ·op 𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) + ((𝐴 ·op 𝑇)‘𝑧))))
414, 39, 40sylanbrc 586 1 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  wf 6324  cfv 6328  (class class class)co 7139  cc 10528  chba 28706   + cva 28707   · csm 28708   ·op chot 28726  LinOpclo 28734
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-mulcom 10594  ax-hilex 28786  ax-hfvadd 28787  ax-hfvmul 28792  ax-hvmulass 28794  ax-hvdistr1 28795
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-homul 29518  df-lnop 29628
This theorem is referenced by:  lnophdi  29789  bdophmi  29819
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