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Theorem lnopcoi 28711
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 28677 . . 3 𝑆: ℋ⟶ ℋ
3 lnopco.2 . . . 4 𝑇 ∈ LinOp
43lnopfi 28677 . . 3 𝑇: ℋ⟶ ℋ
52, 4hocofi 28474 . 2 (𝑆𝑇): ℋ⟶ ℋ
63lnopli 28676 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
76fveq2d 6152 . . . . . . 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 6314 . . . . . . . 8 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
104ffvelrni 6314 . . . . . . . 8 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
111lnopli 28676 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
128, 9, 10, 11syl3an 1365 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
137, 12eqtrd 2655 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
14133expa 1262 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
15 hvmulcl 27719 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
16 hvaddcl 27718 . . . . . . 7 (((𝑥 · 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
1715, 16sylan 488 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) + 𝑧) ∈ ℋ)
182, 4hocoi 28472 . . . . . 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 28472 . . . . . . . 8 (𝑦 ∈ ℋ → ((𝑆𝑇)‘𝑦) = (𝑆‘(𝑇𝑦)))
2120oveq2d 6620 . . . . . . 7 (𝑦 ∈ ℋ → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
2221adantl 482 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
232, 4hocoi 28472 . . . . . 6 (𝑧 ∈ ℋ → ((𝑆𝑇)‘𝑧) = (𝑆‘(𝑇𝑧)))
2422, 23oveqan12d 6623 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
2514, 19, 243eqtr4d 2665 . . . 4 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
26253impa 1256 . . 3 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
2726rgen3 2970 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))
28 ellnop 28566 . 2 ((𝑆𝑇) ∈ LinOp ↔ ((𝑆𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))))
295, 27, 28mpbir2an 954 1 (𝑆𝑇) ∈ LinOp
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  ccom 5078  wf 5843  cfv 5847  (class class class)co 6604  cc 9878  chil 27625   + cva 27626   · csm 27627  LinOpclo 27653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-hilex 27705  ax-hfvadd 27706  ax-hfvmul 27711
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-lnop 28549
This theorem is referenced by:  lnopco0i  28712  nmopcoi  28803  bdopcoi  28806  nmopcoadj0i  28811
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