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Theorem lnocoi 30010
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 = (π‘ˆ LnOp π‘Š)
lnocoi.m 𝑀 = (π‘Š LnOp 𝑋)
lnocoi.n 𝑁 = (π‘ˆ LnOp 𝑋)
lnocoi.u π‘ˆ ∈ NrmCVec
lnocoi.w π‘Š ∈ NrmCVec
lnocoi.x 𝑋 ∈ NrmCVec
lnocoi.s 𝑆 ∈ 𝐿
lnocoi.t 𝑇 ∈ 𝑀
Assertion
Ref Expression
lnocoi (𝑇 ∘ 𝑆) ∈ 𝑁
Proof of Theorem lnocoi
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnocoi.w . . . 4 π‘Š ∈ NrmCVec
2 lnocoi.x . . . 4 𝑋 ∈ NrmCVec
3 lnocoi.t . . . 4 𝑇 ∈ 𝑀
4 eqid 2733 . . . . 5 (BaseSetβ€˜π‘Š) = (BaseSetβ€˜π‘Š)
5 eqid 2733 . . . . 5 (BaseSetβ€˜π‘‹) = (BaseSetβ€˜π‘‹)
6 lnocoi.m . . . . 5 𝑀 = (π‘Š LnOp 𝑋)
74, 5, 6lnof 30008 . . . 4 ((π‘Š ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇 ∈ 𝑀) β†’ 𝑇:(BaseSetβ€˜π‘Š)⟢(BaseSetβ€˜π‘‹))
81, 2, 3, 7mp3an 1462 . . 3 𝑇:(BaseSetβ€˜π‘Š)⟢(BaseSetβ€˜π‘‹)
9 lnocoi.u . . . 4 π‘ˆ ∈ NrmCVec
10 lnocoi.s . . . 4 𝑆 ∈ 𝐿
11 eqid 2733 . . . . 5 (BaseSetβ€˜π‘ˆ) = (BaseSetβ€˜π‘ˆ)
12 lnocoi.l . . . . 5 𝐿 = (π‘ˆ LnOp π‘Š)
1311, 4, 12lnof 30008 . . . 4 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑆 ∈ 𝐿) β†’ 𝑆:(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘Š))
149, 1, 10, 13mp3an 1462 . . 3 𝑆:(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘Š)
15 fco 6742 . . 3 ((𝑇:(BaseSetβ€˜π‘Š)⟢(BaseSetβ€˜π‘‹) ∧ 𝑆:(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘Š)) β†’ (𝑇 ∘ 𝑆):(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘‹))
168, 14, 15mp2an 691 . 2 (𝑇 ∘ 𝑆):(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘‹)
17 eqid 2733 . . . . . . . 8 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
1811, 17nvscl 29879 . . . . . . 7 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ)) β†’ (π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦) ∈ (BaseSetβ€˜π‘ˆ))
199, 18mp3an1 1449 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ)) β†’ (π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦) ∈ (BaseSetβ€˜π‘ˆ))
20 eqid 2733 . . . . . . . 8 ( +𝑣 β€˜π‘ˆ) = ( +𝑣 β€˜π‘ˆ)
2111, 20nvgcl 29873 . . . . . . 7 ((π‘ˆ ∈ NrmCVec ∧ (π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦) ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧) ∈ (BaseSetβ€˜π‘ˆ))
229, 21mp3an1 1449 . . . . . 6 (((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦) ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧) ∈ (BaseSetβ€˜π‘ˆ))
2319, 22stoic3 1779 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧) ∈ (BaseSetβ€˜π‘ˆ))
24 fvco3 6991 . . . . 5 ((𝑆:(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘Š) ∧ ((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧) ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((𝑇 ∘ 𝑆)β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧)) = (π‘‡β€˜(π‘†β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧))))
2514, 23, 24sylancr 588 . . . 4 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((𝑇 ∘ 𝑆)β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧)) = (π‘‡β€˜(π‘†β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧))))
26 id 22 . . . . . 6 (π‘₯ ∈ β„‚ β†’ π‘₯ ∈ β„‚)
2714ffvelcdmi 7086 . . . . . 6 (𝑦 ∈ (BaseSetβ€˜π‘ˆ) β†’ (π‘†β€˜π‘¦) ∈ (BaseSetβ€˜π‘Š))
2814ffvelcdmi 7086 . . . . . 6 (𝑧 ∈ (BaseSetβ€˜π‘ˆ) β†’ (π‘†β€˜π‘§) ∈ (BaseSetβ€˜π‘Š))
291, 2, 33pm3.2i 1340 . . . . . . 7 (π‘Š ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇 ∈ 𝑀)
30 eqid 2733 . . . . . . . 8 ( +𝑣 β€˜π‘Š) = ( +𝑣 β€˜π‘Š)
31 eqid 2733 . . . . . . . 8 ( +𝑣 β€˜π‘‹) = ( +𝑣 β€˜π‘‹)
32 eqid 2733 . . . . . . . 8 ( ·𝑠OLD β€˜π‘Š) = ( ·𝑠OLD β€˜π‘Š)
33 eqid 2733 . . . . . . . 8 ( ·𝑠OLD β€˜π‘‹) = ( ·𝑠OLD β€˜π‘‹)
344, 5, 30, 31, 32, 33, 6lnolin 30007 . . . . . . 7 (((π‘Š ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec ∧ 𝑇 ∈ 𝑀) ∧ (π‘₯ ∈ β„‚ ∧ (π‘†β€˜π‘¦) ∈ (BaseSetβ€˜π‘Š) ∧ (π‘†β€˜π‘§) ∈ (BaseSetβ€˜π‘Š))) β†’ (π‘‡β€˜((π‘₯( ·𝑠OLD β€˜π‘Š)(π‘†β€˜π‘¦))( +𝑣 β€˜π‘Š)(π‘†β€˜π‘§))) = ((π‘₯( ·𝑠OLD β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘¦)))( +𝑣 β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘§))))
3529, 34mpan 689 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ (π‘†β€˜π‘¦) ∈ (BaseSetβ€˜π‘Š) ∧ (π‘†β€˜π‘§) ∈ (BaseSetβ€˜π‘Š)) β†’ (π‘‡β€˜((π‘₯( ·𝑠OLD β€˜π‘Š)(π‘†β€˜π‘¦))( +𝑣 β€˜π‘Š)(π‘†β€˜π‘§))) = ((π‘₯( ·𝑠OLD β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘¦)))( +𝑣 β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘§))))
3626, 27, 28, 35syl3an 1161 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ (π‘‡β€˜((π‘₯( ·𝑠OLD β€˜π‘Š)(π‘†β€˜π‘¦))( +𝑣 β€˜π‘Š)(π‘†β€˜π‘§))) = ((π‘₯( ·𝑠OLD β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘¦)))( +𝑣 β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘§))))
379, 1, 103pm3.2i 1340 . . . . . . 7 (π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑆 ∈ 𝐿)
3811, 4, 20, 30, 17, 32, 12lnolin 30007 . . . . . . 7 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑆 ∈ 𝐿) ∧ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ))) β†’ (π‘†β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘Š)(π‘†β€˜π‘¦))( +𝑣 β€˜π‘Š)(π‘†β€˜π‘§)))
3937, 38mpan 689 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ (π‘†β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘Š)(π‘†β€˜π‘¦))( +𝑣 β€˜π‘Š)(π‘†β€˜π‘§)))
4039fveq2d 6896 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ (π‘‡β€˜(π‘†β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧))) = (π‘‡β€˜((π‘₯( ·𝑠OLD β€˜π‘Š)(π‘†β€˜π‘¦))( +𝑣 β€˜π‘Š)(π‘†β€˜π‘§))))
41 simp2 1138 . . . . . . . 8 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ 𝑦 ∈ (BaseSetβ€˜π‘ˆ))
42 fvco3 6991 . . . . . . . 8 ((𝑆:(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘Š) ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((𝑇 ∘ 𝑆)β€˜π‘¦) = (π‘‡β€˜(π‘†β€˜π‘¦)))
4314, 41, 42sylancr 588 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((𝑇 ∘ 𝑆)β€˜π‘¦) = (π‘‡β€˜(π‘†β€˜π‘¦)))
4443oveq2d 7425 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ (π‘₯( ·𝑠OLD β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘¦)) = (π‘₯( ·𝑠OLD β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘¦))))
45 simp3 1139 . . . . . . 7 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ 𝑧 ∈ (BaseSetβ€˜π‘ˆ))
46 fvco3 6991 . . . . . . 7 ((𝑆:(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘Š) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((𝑇 ∘ 𝑆)β€˜π‘§) = (π‘‡β€˜(π‘†β€˜π‘§)))
4714, 45, 46sylancr 588 . . . . . 6 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((𝑇 ∘ 𝑆)β€˜π‘§) = (π‘‡β€˜(π‘†β€˜π‘§)))
4844, 47oveq12d 7427 . . . . 5 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((π‘₯( ·𝑠OLD β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘¦))( +𝑣 β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘§)) = ((π‘₯( ·𝑠OLD β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘¦)))( +𝑣 β€˜π‘‹)(π‘‡β€˜(π‘†β€˜π‘§))))
4936, 40, 483eqtr4rd 2784 . . . 4 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((π‘₯( ·𝑠OLD β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘¦))( +𝑣 β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘§)) = (π‘‡β€˜(π‘†β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧))))
5025, 49eqtr4d 2776 . . 3 ((π‘₯ ∈ β„‚ ∧ 𝑦 ∈ (BaseSetβ€˜π‘ˆ) ∧ 𝑧 ∈ (BaseSetβ€˜π‘ˆ)) β†’ ((𝑇 ∘ 𝑆)β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘¦))( +𝑣 β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘§)))
5150rgen3 3203 . 2 βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (BaseSetβ€˜π‘ˆ)βˆ€π‘§ ∈ (BaseSetβ€˜π‘ˆ)((𝑇 ∘ 𝑆)β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘¦))( +𝑣 β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘§))
52 lnocoi.n . . . 4 𝑁 = (π‘ˆ LnOp 𝑋)
5311, 5, 20, 31, 17, 33, 52islno 30006 . . 3 ((π‘ˆ ∈ NrmCVec ∧ 𝑋 ∈ NrmCVec) β†’ ((𝑇 ∘ 𝑆) ∈ 𝑁 ↔ ((𝑇 ∘ 𝑆):(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (BaseSetβ€˜π‘ˆ)βˆ€π‘§ ∈ (BaseSetβ€˜π‘ˆ)((𝑇 ∘ 𝑆)β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘¦))( +𝑣 β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘§)))))
549, 2, 53mp2an 691 . 2 ((𝑇 ∘ 𝑆) ∈ 𝑁 ↔ ((𝑇 ∘ 𝑆):(BaseSetβ€˜π‘ˆ)⟢(BaseSetβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ β„‚ βˆ€π‘¦ ∈ (BaseSetβ€˜π‘ˆ)βˆ€π‘§ ∈ (BaseSetβ€˜π‘ˆ)((𝑇 ∘ 𝑆)β€˜((π‘₯( ·𝑠OLD β€˜π‘ˆ)𝑦)( +𝑣 β€˜π‘ˆ)𝑧)) = ((π‘₯( ·𝑠OLD β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘¦))( +𝑣 β€˜π‘‹)((𝑇 ∘ 𝑆)β€˜π‘§))))
5516, 51, 54mpbir2an 710 1 (𝑇 ∘ 𝑆) ∈ 𝑁
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  NrmCVeccnv 29837   +𝑣 cpv 29838  BaseSetcba 29839   ·𝑠OLD cns 29840   LnOp clno 29993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-grpo 29746  df-ablo 29798  df-vc 29812  df-nv 29845  df-va 29848  df-ba 29849  df-sm 29850  df-0v 29851  df-nmcv 29853  df-lno 29997
This theorem is referenced by: (None)
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