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Theorem lnopmi 28708
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 28677 . . 3 𝑇: ℋ⟶ ℋ
3 homulcl 28467 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
42, 3mpan2 706 . 2 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5 hvmulcl 27719 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
6 hvaddcl 27718 . . . . . . . 8 (((𝑥 · 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
75, 6sylan 488 . . . . . . 7 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
8 homval 28449 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
92, 8mp3an2 1409 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
107, 9sylan2 491 . . . . . 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 6314 . . . . . . . . . 10 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
13 hvmulcl 27719 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
1412, 13sylan2 491 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
152ffvelrni 6314 . . . . . . . . 9 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
16 ax-hvdistr1 27714 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 · (𝑇𝑦)) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
1711, 14, 15, 16syl3an 1365 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
18173expb 1263 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
191lnopli 28676 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
20193expa 1262 . . . . . . . . 9 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
2120oveq2d 6620 . . . . . . . 8 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
2221adantl 482 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
23 homval 28449 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
242, 23mp3an2 1409 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2524adantrl 751 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2625oveq2d 6620 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 · (𝐴 · (𝑇𝑦))))
27 hvmulcom 27749 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
2812, 27syl3an3 1358 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
29283expb 1263 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
3026, 29eqtr4d 2658 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 · (𝑥 · (𝑇𝑦))))
31 homval 28449 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
322, 31mp3an2 1409 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
3330, 32oveqan12d 6623 . . . . . . . 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 872 . . . . . . 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 2666 . . . . . 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 2658 . . . . 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 630 . . . 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 2962 . . 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 2964 . 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 28566 . 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 697 1 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wf 5843  cfv 5847  (class class class)co 6604  cc 9878  chil 27625   + cva 27626   · csm 27627   ·op chot 27645  LinOpclo 27653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-mulcom 9944  ax-hilex 27705  ax-hfvadd 27706  ax-hfvmul 27711  ax-hvmulass 27713  ax-hvdistr1 27714
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-homul 28439  df-lnop 28549
This theorem is referenced by:  lnophdi  28710  bdophmi  28740
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