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Theorem lnopcoi 29830
 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 29796 . . 3 𝑆: ℋ⟶ ℋ
3 lnopco.2 . . . 4 𝑇 ∈ LinOp
43lnopfi 29796 . . 3 𝑇: ℋ⟶ ℋ
52, 4hocofi 29593 . 2 (𝑆𝑇): ℋ⟶ ℋ
63lnopli 29795 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧)))
76fveq2d 6659 . . . . . . 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 6837 . . . . . . . 8 (𝑦 ∈ ℋ → (𝑇𝑦) ∈ ℋ)
104ffvelrni 6837 . . . . . . . 8 (𝑧 ∈ ℋ → (𝑇𝑧) ∈ ℋ)
111lnopli 29795 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ (𝑇𝑦) ∈ ℋ ∧ (𝑇𝑧) ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
128, 9, 10, 11syl3an 1157 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘((𝑥 · (𝑇𝑦)) + (𝑇𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
137, 12eqtrd 2833 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
14133expa 1115 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝑆‘(𝑇‘((𝑥 · 𝑦) + 𝑧))) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
15 hvmulcl 28840 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
16 hvaddcl 28839 . . . . . . 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 29591 . . . . . 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 29591 . . . . . . . 8 (𝑦 ∈ ℋ → ((𝑆𝑇)‘𝑦) = (𝑆‘(𝑇𝑦)))
2120oveq2d 7161 . . . . . . 7 (𝑦 ∈ ℋ → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
2221adantl 485 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · ((𝑆𝑇)‘𝑦)) = (𝑥 · (𝑆‘(𝑇𝑦))))
232, 4hocoi 29591 . . . . . 6 (𝑧 ∈ ℋ → ((𝑆𝑇)‘𝑧) = (𝑆‘(𝑇𝑧)))
2422, 23oveqan12d 7164 . . . . 5 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)) = ((𝑥 · (𝑆‘(𝑇𝑦))) + (𝑆‘(𝑇𝑧))))
2514, 19, 243eqtr4d 2843 . . . 4 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
26253impa 1107 . . 3 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧)))
2726rgen3 3169 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))
28 ellnop 29685 . 2 ((𝑆𝑇) ∈ LinOp ↔ ((𝑆𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ ((𝑆𝑇)‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · ((𝑆𝑇)‘𝑦)) + ((𝑆𝑇)‘𝑧))))
295, 27, 28mpbir2an 710 1 (𝑆𝑇) ∈ LinOp
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∘ ccom 5527  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  ℂcc 10542   ℋchba 28746   +ℎ cva 28747   ·ℎ csm 28748  LinOpclo 28774 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-hilex 28826  ax-hfvadd 28827  ax-hfvmul 28832 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8409  df-lnop 29668 This theorem is referenced by:  lnopco0i  29831  nmopcoi  29922  bdopcoi  29925  nmopcoadj0i  29930
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