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Theorem lnopcoi 30084
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 30050 . . 3 𝑆: ℋ⟶ ℋ
3 lnopco.2 . . . 4 𝑇 ∈ LinOp
43lnopfi 30050 . . 3 𝑇: ℋ⟶ ℋ
52, 4hocofi 29847 . 2 (𝑆𝑇): ℋ⟶ ℋ
63lnopli 30049 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
76fveq2d 6721 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))
8 id 22 . . . . . . . 8 (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
94ffvelrni 6903 . . . . . . . 8 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
104ffvelrni 6903 . . . . . . . 8 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
111lnopli 30049 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
128, 9, 10, 11syl3an 1162 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
137, 12eqtrd 2777 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
14133expa 1120 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
15 hvmulcl 29094 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
16 hvaddcl 29093 . . . . . . 7 (((𝑥 · 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
1715, 16sylan 583 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
182, 4hocoi 29845 . . . . . 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 29845 . . . . . . . 8 (𝑦 ∈ ℋ → ((𝑆𝑇)‘𝑦) = (𝑆‘(𝑇𝑦)))
2120oveq2d 7229 . . . . . . 7 (𝑦 ∈ ℋ → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
2221adantl 485 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
232, 4hocoi 29845 . . . . . 6 (𝑧 ∈ ℋ → ((𝑆𝑇)‘𝑧) = (𝑆‘(𝑇𝑧)))
2422, 23oveqan12d 7232 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
2514, 19, 243eqtr4d 2787 . . . 4 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
26253impa 1112 . . 3 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
2726rgen3 3125 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))
28 ellnop 29939 . 2 ((𝑆𝑇) ∈ LinOp ↔ ((𝑆𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))))
295, 27, 28mpbir2an 711 1 (𝑆𝑇) ∈ LinOp
Colors of variables: wff setvar class
Syntax hints:  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  ccom 5555  wf 6376  cfv 6380  (class class class)co 7213  cc 10727  chba 29000   + cva 29001   · csm 29002  LinOpclo 29028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-hilex 29080  ax-hfvadd 29081  ax-hfvmul 29086
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-lnop 29922
This theorem is referenced by:  lnopco0i  30085  nmopcoi  30176  bdopcoi  30179  nmopcoadj0i  30184
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