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Theorem lnocoi 28526
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 2819 . . . . 5 (BaseSet‘𝑊) = (BaseSet‘𝑊)
5 eqid 2819 . . . . 5 (BaseSet‘𝑋) = (BaseSet‘𝑋)
6 lnocoi.m . . . . 5 𝑀 = (𝑊 LnOp 𝑋)
74, 5, 6lnof 28524 . . . 4 ((𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇𝑀) → 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋))
81, 2, 3, 7mp3an 1454 . . 3 𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋)
9 lnocoi.u . . . 4 𝑈 ∈ NrmCVec
10 lnocoi.s . . . 4 𝑆𝐿
11 eqid 2819 . . . . 5 (BaseSet‘𝑈) = (BaseSet‘𝑈)
12 lnocoi.l . . . . 5 𝐿 = (𝑈 LnOp 𝑊)
1311, 4, 12lnof 28524 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆𝐿) → 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊))
149, 1, 10, 13mp3an 1454 . . 3 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)
15 fco 6524 . . 3 ((𝑇:(BaseSet‘𝑊)⟶(BaseSet‘𝑋) ∧ 𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → (𝑇𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋))
168, 14, 15mp2an 690 . 2 (𝑇𝑆):(BaseSet‘𝑈)⟶(BaseSet‘𝑋)
17 eqid 2819 . . . . . . . 8 ( ·𝑠OLD𝑈) = ( ·𝑠OLD𝑈)
1811, 17nvscl 28395 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
199, 18mp3an1 1441 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈))
20 eqid 2819 . . . . . . . 8 ( +𝑣𝑈) = ( +𝑣𝑈)
2111, 20nvgcl 28389 . . . . . . 7 ((𝑈 ∈ NrmCVec ∧ (𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
229, 21mp3an1 1441 . . . . . 6 (((𝑥( ·𝑠OLD𝑈)𝑦) ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
2319, 22stoic3 1770 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧) ∈ (BaseSet‘𝑈))
24 fvco3 6753 . . . . 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 589 . . . 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 6843 . . . . . 6 (𝑦 ∈ (BaseSet‘𝑈) → (𝑆𝑦) ∈ (BaseSet‘𝑊))
2814ffvelrni 6843 . . . . . 6 (𝑧 ∈ (BaseSet‘𝑈) → (𝑆𝑧) ∈ (BaseSet‘𝑊))
291, 2, 33pm3.2i 1333 . . . . . . 7 (𝑊 ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇𝑀)
30 eqid 2819 . . . . . . . 8 ( +𝑣𝑊) = ( +𝑣𝑊)
31 eqid 2819 . . . . . . . 8 ( +𝑣𝑋) = ( +𝑣𝑋)
32 eqid 2819 . . . . . . . 8 ( ·𝑠OLD𝑊) = ( ·𝑠OLD𝑊)
33 eqid 2819 . . . . . . . 8 ( ·𝑠OLD𝑋) = ( ·𝑠OLD𝑋)
344, 5, 30, 31, 32, 33, 6lnolin 28523 . . . . . . 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 688 . . . . . 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 1154 . . . . 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 1333 . . . . . . 7 (𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑆𝐿)
3811, 4, 20, 30, 17, 32, 12lnolin 28523 . . . . . . 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 688 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧)))
4039fveq2d 6667 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑇‘(𝑆‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧))) = (𝑇‘((𝑥( ·𝑠OLD𝑊)(𝑆𝑦))( +𝑣𝑊)(𝑆𝑧))))
41 simp2 1131 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑦 ∈ (BaseSet‘𝑈))
42 fvco3 6753 . . . . . . . 8 ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
4314, 41, 42sylancr 589 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑦) = (𝑇‘(𝑆𝑦)))
4443oveq2d 7164 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → (𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦)) = (𝑥( ·𝑠OLD𝑋)(𝑇‘(𝑆𝑦))))
45 simp3 1132 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → 𝑧 ∈ (BaseSet‘𝑈))
46 fvco3 6753 . . . . . . 7 ((𝑆:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑧) = (𝑇‘(𝑆𝑧)))
4714, 45, 46sylancr 589 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘𝑧) = (𝑇‘(𝑆𝑧)))
4844, 47oveq12d 7166 . . . . 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 2865 . . . 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 2857 . . 3 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (BaseSet‘𝑈) ∧ 𝑧 ∈ (BaseSet‘𝑈)) → ((𝑇𝑆)‘((𝑥( ·𝑠OLD𝑈)𝑦)( +𝑣𝑈)𝑧)) = ((𝑥( ·𝑠OLD𝑋)((𝑇𝑆)‘𝑦))( +𝑣𝑋)((𝑇𝑆)‘𝑧)))
5150rgen3 3202 . 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 28522 . . 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 690 . 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 709 1 (𝑇𝑆) ∈ 𝑁
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  ccom 5552  wf 6344  cfv 6348  (class class class)co 7148  cc 10527  NrmCVeccnv 28353   +𝑣 cpv 28354  BaseSetcba 28355   ·𝑠OLD cns 28356   LnOp clno 28509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400  df-grpo 28262  df-ablo 28314  df-vc 28328  df-nv 28361  df-va 28364  df-ba 28365  df-sm 28366  df-0v 28367  df-nmcv 28369  df-lno 28513
This theorem is referenced by: (None)
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