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Theorem lnopmi 30942
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 30911 . . 3 𝑇: ℋ⟶ ℋ
3 homulcl 30701 . . 3 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
42, 3mpan2 689 . 2 (𝐴 ∈ ℂ → (𝐴 ·op 𝑇): ℋ⟶ ℋ)
5 hvmulcl 29955 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
6 hvaddcl 29954 . . . . . . . 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 30683 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
92, 8mp3an2 1449 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 · 𝑦) + 𝑧) ∈ ℋ) → ((𝐴 ·op 𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝐴 · (𝑇‘((𝑥 · 𝑦) + 𝑧))))
107, 9sylan2 593 . . . . . 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)
122ffvelcdmi 7034 . . . . . . . . . 10 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
13 hvmulcl 29955 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
1412, 13sylan2 593 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (𝑇𝑦)) ∈ ℋ)
152ffvelcdmi 7034 . . . . . . . . 9 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
16 ax-hvdistr1 29950 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 · (𝑇𝑦)) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
1711, 14, 15, 16syl3an 1160 . . . . . . . 8 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
18173expb 1120 . . . . . . 7 ((𝐴 ∈ ℂ ∧ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ)) → (𝐴 · ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝐴 · (𝑥 · (𝑇𝑦))) + (𝐴 · (𝑇𝑧))))
191lnopli 30910 . . . . . . . . . 10 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
20193expa 1118 . . . . . . . . 9 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
2120oveq2d 7373 . . . . . . . 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 30683 . . . . . . . . . . . . 13 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
242, 23mp3an2 1449 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2524adantrl 714 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝐴 ·op 𝑇)‘𝑦) = (𝐴 · (𝑇𝑦)))
2625oveq2d 7373 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝑥 · (𝐴 · (𝑇𝑦))))
27 hvmulcom 29985 . . . . . . . . . . . 12 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
2812, 27syl3an3 1165 . . . . . . . . . . 11 ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
29283expb 1120 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝐴 · (𝑥 · (𝑇𝑦))) = (𝑥 · (𝐴 · (𝑇𝑦))))
3026, 29eqtr4d 2779 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (𝑥 · ((𝐴 ·op 𝑇)‘𝑦)) = (𝐴 · (𝑥 · (𝑇𝑦))))
31 homval 30683 . . . . . . . . . 10 ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
322, 31mp3an2 1449 . . . . . . . . 9 ((𝐴 ∈ ℂ ∧ 𝑧 ∈ ℋ) → ((𝐴 ·op 𝑇)‘𝑧) = (𝐴 · (𝑇𝑧)))
3330, 32oveqan12d 7376 . . . . . . . 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 676 . . . . . . 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 2787 . . . . . 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 2779 . . . . 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 3149 . . 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 3195 . 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 30800 . 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 1541  wcel 2106  wral 3064  wf 6492  cfv 6496  (class class class)co 7357  cc 11049  chba 29861   + cva 29862   · csm 29863   ·op chot 29881  LinOpclo 29889
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-mulcom 11115  ax-hilex 29941  ax-hfvadd 29942  ax-hfvmul 29947  ax-hvmulass 29949  ax-hvdistr1 29950
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-homul 30673  df-lnop 30783
This theorem is referenced by:  lnophdi  30944  bdophmi  30974
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