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Theorem lnopmi 28987
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 28956 . . 3 𝑇: ℋ⟶ ℋ
3 homulcl 28746 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
42, 3mpan2 707 . 2 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5 hvmulcl 27998 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
6 hvaddcl 27997 . . . . . . . 8 (((𝑥 · 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
75, 6sylan 487 . . . . . . 7 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
8 homval 28728 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
92, 8mp3an2 1452 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
107, 9sylan2 490 . . . . . 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 6398 . . . . . . . . . 10 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
13 hvmulcl 27998 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
1412, 13sylan2 490 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
152ffvelrni 6398 . . . . . . . . 9 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
16 ax-hvdistr1 27993 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 · (𝑇𝑦)) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
1711, 14, 15, 16syl3an 1408 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
18173expb 1285 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
191lnopli 28955 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
20193expa 1284 . . . . . . . . 9 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
2120oveq2d 6706 . . . . . . . 8 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
2221adantl 481 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
23 homval 28728 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
242, 23mp3an2 1452 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2524adantrl 752 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2625oveq2d 6706 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 · (𝐴 · (𝑇𝑦))))
27 hvmulcom 28028 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
2812, 27syl3an3 1401 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
29283expb 1285 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
3026, 29eqtr4d 2688 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 · (𝑥 · (𝑇𝑦))))
31 homval 28728 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
322, 31mp3an2 1452 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
3330, 32oveqan12d 6709 . . . . . . . 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 890 . . . . . . 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 2696 . . . . . 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 2688 . . . . 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 2997 . . 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 2999 . 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 28845 . 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 699 1 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wf 5922  cfv 5926  (class class class)co 6690  cc 9972  chil 27904   + cva 27905   · csm 27906   ·op chot 27924  LinOpclo 27932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-mulcom 10038  ax-hilex 27984  ax-hfvadd 27985  ax-hfvmul 27990  ax-hvmulass 27992  ax-hvdistr1 27993
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-homul 28718  df-lnop 28828
This theorem is referenced by:  lnophdi  28989  bdophmi  29019
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