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Theorem lnopmi 29704
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 29673 . . 3 𝑇: ℋ⟶ ℋ
3 homulcl 29463 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
42, 3mpan2 687 . 2 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5 hvmulcl 28717 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
6 hvaddcl 28716 . . . . . . . 8 (((𝑥 · 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
75, 6sylan 580 . . . . . . 7 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
8 homval 29445 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
92, 8mp3an2 1440 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
107, 9sylan2 592 . . . . . 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 6842 . . . . . . . . . 10 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
13 hvmulcl 28717 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
1412, 13sylan2 592 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
152ffvelrni 6842 . . . . . . . . 9 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
16 ax-hvdistr1 28712 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 · (𝑇𝑦)) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
1711, 14, 15, 16syl3an 1152 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
18173expb 1112 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
191lnopli 29672 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
20193expa 1110 . . . . . . . . 9 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
2120oveq2d 7161 . . . . . . . 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 29445 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
242, 23mp3an2 1440 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2524adantrl 712 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2625oveq2d 7161 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 · (𝐴 · (𝑇𝑦))))
27 hvmulcom 28747 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
2812, 27syl3an3 1157 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
29283expb 1112 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
3026, 29eqtr4d 2856 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 · (𝑥 · (𝑇𝑦))))
31 homval 29445 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
322, 31mp3an2 1440 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
3330, 32oveqan12d 7164 . . . . . . . 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 674 . . . . . . 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 2864 . . . . . 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 2856 . . . . 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 421 . . . 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 3185 . . 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 3187 . 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 29562 . 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 583 1 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇) ∈ LinOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  chba 28623   + cva 28624   · csm 28625   ·op chot 28643  LinOpclo 28651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-mulcom 10589  ax-hilex 28703  ax-hfvadd 28704  ax-hfvmul 28709  ax-hvmulass 28711  ax-hvdistr1 28712
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-homul 29435  df-lnop 29545
This theorem is referenced by:  lnophdi  29706  bdophmi  29736
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