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Theorem lnocoi 27946
Description: The composition of two linear operators is linear. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnocoi.l 𝐿 = (𝑈 LnOp 𝑊)
lnocoi.m 𝑀 = (𝑊 LnOp 𝑋)
lnocoi.n 𝑁 = (𝑈 LnOp 𝑋)
lnocoi.u 𝑈 ∈ NrmCVec
lnocoi.w 𝑊 ∈ NrmCVec
lnocoi.x 𝑋 ∈ NrmCVec
lnocoi.s 𝑆𝐿
lnocoi.t 𝑇𝑀
Assertion
Ref Expression
lnocoi (𝑇𝑆) ∈ 𝑁

Proof of Theorem lnocoi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnocoi.w . . . 4 𝑊 ∈ NrmCVec
2 lnocoi.x . . . 4 𝑋 ∈ NrmCVec
3 lnocoi.t . . . 4 𝑇𝑀
4 eqid 2770 . . . . 5 (BaseSet‘𝑊) = (BaseSet‘𝑊)
5 eqid 2770 . . . . 5 (BaseSet‘𝑋) = (BaseSet‘𝑋)
6 lnocoi.m . . . . 5 𝑀 = (𝑊 LnOp 𝑋)
74, 5, 6lnof 27944 . . . 4 ((𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇𝑀) → 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋))
81, 2, 3, 7mp3an 1571 . . 3 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋)
9 lnocoi.u . . . 4 𝑈 ∈ NrmCVec
10 lnocoi.s . . . 4 𝑆𝐿
11 eqid 2770 . . . . 5 (BaseSet‘𝑈) = (BaseSet‘𝑈)
12 lnocoi.l . . . . 5 𝐿 = (𝑈 LnOp 𝑊)
1311, 4, 12lnof 27944 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆𝐿) → 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
149, 1, 10, 13mp3an 1571 . . 3 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
15 fco 6198 . . 3 ((𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋) ∧ 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑇𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋))
168, 14, 15mp2an 664 . 2 (𝑇𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋)
17 eqid 2770 . . . . . . . 8 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
1811, 17nvscl 27815 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
199, 18mp3an1 1558 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
20 eqid 2770 . . . . . . . 8 ( +𝑣𝑈) = ( +𝑣𝑈)
2111, 20nvgcl 27809 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
229, 21mp3an1 1558 . . . . . 6 (((𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
2319, 22stoic3 1848 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
24 fvco3 6417 . . . . 5 ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = (𝑇‘(𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧))))
2514, 23, 24sylancr 567 . . . 4 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = (𝑇‘(𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧))))
26 id 22 . . . . . 6 (𝑥 ∈ ℂ → 𝑥 ∈ ℂ)
2714ffvelrni 6501 . . . . . 6 (𝑦 ∈ (BaseSet‘𝑈) → (𝑆𝑦) ∈ (BaseSet‘𝑊))
2814ffvelrni 6501 . . . . . 6 (𝑧 ∈ (BaseSet‘𝑈) → (𝑆𝑧) ∈ (BaseSet‘𝑊))
291, 2, 33pm3.2i 1422 . . . . . . 7 (𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇𝑀)
30 eqid 2770 . . . . . . . 8 ( +𝑣𝑊) = ( +𝑣𝑊)
31 eqid 2770 . . . . . . . 8 ( +𝑣𝑋) = ( +𝑣𝑋)
32 eqid 2770 . . . . . . . 8 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
33 eqid 2770 . . . . . . . 8 ( ·𝑠OLD𝑋) = ( ·𝑠OLD𝑋)
344, 5, 30, 31, 32, 33, 6lnolin 27943 . . . . . . 7 (((𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇𝑀) ∧ (𝑥 ∈ ℂ ∧ (𝑆𝑦) ∈ (BaseSet‘𝑊) ∧ (𝑆𝑧) ∈ (BaseSet‘𝑊))) → (𝑇‘((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧))) = ((𝑥( ·𝑠OLD𝑋)(𝑇‘(𝑆𝑦)))( +𝑣𝑋)(𝑇‘(𝑆𝑧))))
3529, 34mpan 662 . . . . . 6 ((𝑥 ∈ ℂ ∧ (𝑆𝑦) ∈ (BaseSet‘𝑊) ∧ (𝑆𝑧) ∈ (BaseSet‘𝑊)) → (𝑇‘((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧))) = ((𝑥( ·𝑠OLD𝑋)(𝑇‘(𝑆𝑦)))( +𝑣𝑋)(𝑇‘(𝑆𝑧))))
3626, 27, 28, 35syl3an 1162 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑇‘((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧))) = ((𝑥( ·𝑠OLD𝑋)(𝑇‘(𝑆𝑦)))( +𝑣𝑋)(𝑇‘(𝑆𝑧))))
379, 1, 103pm3.2i 1422 . . . . . . 7 (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆𝐿)
3811, 4, 20, 30, 17, 32, 12lnolin 27943 . . . . . . 7 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆𝐿) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈))) → (𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧)))
3937, 38mpan 662 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧)))
4039fveq2d 6336 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑇‘(𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧))) = (𝑇‘((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧))))
41 simp2 1130 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ (BaseSet‘𝑈))
42 fvco3 6417 . . . . . . . 8 ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
4314, 41, 42sylancr 567 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
4443oveq2d 6808 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦)) = (𝑥( ·𝑠OLD𝑋)(𝑇‘(𝑆𝑦))))
45 simp3 1131 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑧 ∈ (BaseSet‘𝑈))
46 fvco3 6417 . . . . . . 7 ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑧) = (𝑇‘(𝑆𝑧)))
4714, 45, 46sylancr 567 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑧) = (𝑇‘(𝑆𝑧)))
4844, 47oveq12d 6810 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦))( +𝑣𝑋)((𝑇𝑆)‘𝑧)) = ((𝑥( ·𝑠OLD𝑋)(𝑇‘(𝑆𝑦)))( +𝑣𝑋)(𝑇‘(𝑆𝑧))))
4936, 40, 483eqtr4rd 2815 . . . 4 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦))( +𝑣𝑋)((𝑇𝑆)‘𝑧)) = (𝑇‘(𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧))))
5025, 49eqtr4d 2807 . . 3 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦))( +𝑣𝑋)((𝑇𝑆)‘𝑧)))
5150rgen3 3124 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)((𝑇𝑆)‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦))( +𝑣𝑋)((𝑇𝑆)‘𝑧))
52 lnocoi.n . . . 4 𝑁 = (𝑈 LnOp 𝑋)
5311, 5, 20, 31, 17, 33, 52islno 27942 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec) → ((𝑇𝑆) ∈ 𝑁 ↔ ((𝑇𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)((𝑇𝑆)‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦))( +𝑣𝑋)((𝑇𝑆)‘𝑧)))))
549, 2, 53mp2an 664 . 2 ((𝑇𝑆) ∈ 𝑁 ↔ ((𝑇𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (BaseSet‘𝑈)∀𝑧 ∈ (BaseSet‘𝑈)((𝑇𝑆)‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦))( +𝑣𝑋)((𝑇𝑆)‘𝑧))))
5516, 51, 54mpbir2an 682 1 (𝑇𝑆) ∈ 𝑁
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  ccom 5253  wf 6027  cfv 6031  (class class class)co 6792  cc 10135  NrmCVeccnv 27773   +𝑣 cpv 27774  BaseSetcba 27775   ·𝑠OLD cns 27776   LnOp clno 27929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010  df-grpo 27681  df-ablo 27733  df-vc 27748  df-nv 27781  df-va 27784  df-ba 27785  df-sm 27786  df-0v 27787  df-nmcv 27789  df-lno 27933
This theorem is referenced by: (None)
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