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Theorem lnocoi 27596
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 2621 . . . . 5 (BaseSet‘𝑊) = (BaseSet‘𝑊)
5 eqid 2621 . . . . 5 (BaseSet‘𝑋) = (BaseSet‘𝑋)
6 lnocoi.m . . . . 5 𝑀 = (𝑊 LnOp 𝑋)
74, 5, 6lnof 27594 . . . 4 ((𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇𝑀) → 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋))
81, 2, 3, 7mp3an 1423 . . 3 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋)
9 lnocoi.u . . . 4 𝑈 ∈ NrmCVec
10 lnocoi.s . . . 4 𝑆𝐿
11 eqid 2621 . . . . 5 (BaseSet‘𝑈) = (BaseSet‘𝑈)
12 lnocoi.l . . . . 5 𝐿 = (𝑈 LnOp 𝑊)
1311, 4, 12lnof 27594 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆𝐿) → 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
149, 1, 10, 13mp3an 1423 . . 3 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
15 fco 6056 . . 3 ((𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋) ∧ 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑇𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋))
168, 14, 15mp2an 708 . 2 (𝑇𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋)
17 eqid 2621 . . . . . . . 8 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
1811, 17nvscl 27465 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
199, 18mp3an1 1410 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
20 eqid 2621 . . . . . . . 8 ( +𝑣𝑈) = ( +𝑣𝑈)
2111, 20nvgcl 27459 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
229, 21mp3an1 1410 . . . . . 6 (((𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
2319, 22stoic3 1700 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
24 fvco3 6273 . . . . 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 695 . . . 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 6356 . . . . . 6 (𝑦 ∈ (BaseSet‘𝑈) → (𝑆𝑦) ∈ (BaseSet‘𝑊))
2814ffvelrni 6356 . . . . . 6 (𝑧 ∈ (BaseSet‘𝑈) → (𝑆𝑧) ∈ (BaseSet‘𝑊))
291, 2, 33pm3.2i 1238 . . . . . . 7 (𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇𝑀)
30 eqid 2621 . . . . . . . 8 ( +𝑣𝑊) = ( +𝑣𝑊)
31 eqid 2621 . . . . . . . 8 ( +𝑣𝑋) = ( +𝑣𝑋)
32 eqid 2621 . . . . . . . 8 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
33 eqid 2621 . . . . . . . 8 ( ·𝑠OLD𝑋) = ( ·𝑠OLD𝑋)
344, 5, 30, 31, 32, 33, 6lnolin 27593 . . . . . . 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 706 . . . . . 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 1367 . . . . 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 1238 . . . . . . 7 (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆𝐿)
3811, 4, 20, 30, 17, 32, 12lnolin 27593 . . . . . . 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 706 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧)))
4039fveq2d 6193 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑇‘(𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧))) = (𝑇‘((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧))))
41 simp2 1061 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ (BaseSet‘𝑈))
42 fvco3 6273 . . . . . . . 8 ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
4314, 41, 42sylancr 695 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
4443oveq2d 6663 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦)) = (𝑥( ·𝑠OLD𝑋)(𝑇‘(𝑆𝑦))))
45 simp3 1062 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑧 ∈ (BaseSet‘𝑈))
46 fvco3 6273 . . . . . . 7 ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑧) = (𝑇‘(𝑆𝑧)))
4714, 45, 46sylancr 695 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑧) = (𝑇‘(𝑆𝑧)))
4844, 47oveq12d 6665 . . . . 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 2666 . . . 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 2658 . . 3 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦))( +𝑣𝑋)((𝑇𝑆)‘𝑧)))
5150rgen3 2975 . 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 27592 . . 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 708 . 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 955 1 (𝑇𝑆) ∈ 𝑁
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  ccom 5116  wf 5882  cfv 5886  (class class class)co 6647  cc 9931  NrmCVeccnv 27423   +𝑣 cpv 27424  BaseSetcba 27425   ·𝑠OLD cns 27426   LnOp clno 27579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-map 7856  df-grpo 27331  df-ablo 27383  df-vc 27398  df-nv 27431  df-va 27434  df-ba 27435  df-sm 27436  df-0v 27437  df-nmcv 27439  df-lno 27583
This theorem is referenced by: (None)
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