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Theorem lnopcoi 30653
Description: The composition of two linear operators is linear. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnopco.1 𝑆 ∈ LinOp
lnopco.2 𝑇 ∈ LinOp
Assertion
Ref Expression
lnopcoi (𝑆𝑇) ∈ LinOp

Proof of Theorem lnopcoi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnopco.1 . . . 4 𝑆 ∈ LinOp
21lnopfi 30619 . . 3 𝑆: ℋ⟶ ℋ
3 lnopco.2 . . . 4 𝑇 ∈ LinOp
43lnopfi 30619 . . 3 𝑇: ℋ⟶ ℋ
52, 4hocofi 30416 . 2 (𝑆𝑇): ℋ⟶ ℋ
63lnopli 30618 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
76fveq2d 6829 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
8 id 22 . . . . . . . 8 (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
94ffvelcdmi 7016 . . . . . . . 8 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
104ffvelcdmi 7016 . . . . . . . 8 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
111lnopli 30618 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
128, 9, 10, 11syl3an 1159 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
137, 12eqtrd 2776 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
14133expa 1117 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
15 hvmulcl 29663 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
16 hvaddcl 29662 . . . . . . 7 (((𝑥 · 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
1715, 16sylan 580 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
182, 4hocoi 30414 . . . . . 6 (((𝑥 · 𝑦) + 𝑧) ∈ ℋ → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))))
1917, 18syl 17 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))))
202, 4hocoi 30414 . . . . . . . 8 (𝑦 ∈ ℋ → ((𝑆𝑇)‘𝑦) = (𝑆‘(𝑇𝑦)))
2120oveq2d 7353 . . . . . . 7 (𝑦 ∈ ℋ → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
2221adantl 482 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
232, 4hocoi 30414 . . . . . 6 (𝑧 ∈ ℋ → ((𝑆𝑇)‘𝑧) = (𝑆‘(𝑇𝑧)))
2422, 23oveqan12d 7356 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
2514, 19, 243eqtr4d 2786 . . . 4 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
26253impa 1109 . . 3 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
2726rgen3 3195 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))
28 ellnop 30508 . 2 ((𝑆𝑇) ∈ LinOp ↔ ((𝑆𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))))
295, 27, 28mpbir2an 708 1 (𝑆𝑇) ∈ LinOp
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3061  ccom 5624  wf 6475  cfv 6479  (class class class)co 7337  cc 10970  chba 29569   + cva 29570   · csm 29571  LinOpclo 29597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-hilex 29649  ax-hfvadd 29650  ax-hfvmul 29655
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-map 8688  df-lnop 30491
This theorem is referenced by:  lnopco0i  30654  nmopcoi  30745  bdopcoi  30748  nmopcoadj0i  30753
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