MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmhmco Structured version   Visualization version   GIF version

Theorem lmhmco 20350
Description: The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Assertion
Ref Expression
lmhmco ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘ 𝐺) ∈ (𝑀 LMHom 𝑂))

Proof of Theorem lmhmco
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . 2 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2736 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 eqid 2736 . 2 ( ·𝑠 β€˜π‘‚) = ( ·𝑠 β€˜π‘‚)
4 eqid 2736 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5 eqid 2736 . 2 (Scalarβ€˜π‘‚) = (Scalarβ€˜π‘‚)
6 eqid 2736 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
7 lmhmlmod1 20340 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
87adantl 483 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20339 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) β†’ 𝑂 ∈ LMod)
109adantr 482 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑂 ∈ LMod)
11 eqid 2736 . . . 4 (Scalarβ€˜π‘) = (Scalarβ€˜π‘)
1211, 5lmhmsca 20337 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) β†’ (Scalarβ€˜π‘‚) = (Scalarβ€˜π‘))
134, 11lmhmsca 20337 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ (Scalarβ€˜π‘) = (Scalarβ€˜π‘€))
1412, 13sylan9eq 2796 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (Scalarβ€˜π‘‚) = (Scalarβ€˜π‘€))
15 lmghm 20338 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) β†’ 𝐹 ∈ (𝑁 GrpHom 𝑂))
16 lmghm 20338 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺 ∈ (𝑀 GrpHom 𝑁))
17 ghmco 18899 . . 3 ((𝐹 ∈ (𝑁 GrpHom 𝑂) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) β†’ (𝐹 ∘ 𝐺) ∈ (𝑀 GrpHom 𝑂))
1815, 16, 17syl2an 597 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘ 𝐺) ∈ (𝑀 GrpHom 𝑂))
19 simplr 767 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺 ∈ (𝑀 LMHom 𝑁))
20 simprl 769 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
21 simprr 771 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑦 ∈ (Baseβ€˜π‘€))
22 eqid 2736 . . . . . . 7 ( ·𝑠 β€˜π‘) = ( ·𝑠 β€˜π‘)
234, 6, 1, 2, 22lmhmlin 20342 . . . . . 6 ((𝐺 ∈ (𝑀 LMHom 𝑁) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦)))
2419, 20, 21, 23syl3anc 1371 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦)))
2524fveq2d 6808 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΉβ€˜(πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦))) = (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦))))
26 simpll 765 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹 ∈ (𝑁 LMHom 𝑂))
2713fveq2d 6808 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘€)))
2827ad2antlr 725 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘€)))
2920, 28eleqtrrd 2840 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘)))
30 eqid 2736 . . . . . . . . 9 (Baseβ€˜π‘) = (Baseβ€˜π‘)
311, 30lmhmf 20341 . . . . . . . 8 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
3231adantl 483 . . . . . . 7 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐺:(Baseβ€˜π‘€)⟢(Baseβ€˜π‘))
3332ffvelcdmda 6993 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘))
3433adantrl 714 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘))
35 eqid 2736 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘))
3611, 35, 30, 22, 3lmhmlin 20342 . . . . 5 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘)) ∧ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘)) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦))) = (π‘₯( ·𝑠 β€˜π‘‚)(πΉβ€˜(πΊβ€˜π‘¦))))
3726, 29, 34, 36syl3anc 1371 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΉβ€˜(π‘₯( ·𝑠 β€˜π‘)(πΊβ€˜π‘¦))) = (π‘₯( ·𝑠 β€˜π‘‚)(πΉβ€˜(πΊβ€˜π‘¦))))
3825, 37eqtrd 2776 . . 3 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΉβ€˜(πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦))) = (π‘₯( ·𝑠 β€˜π‘‚)(πΉβ€˜(πΊβ€˜π‘¦))))
3932ffnd 6631 . . . 4 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐺 Fn (Baseβ€˜π‘€))
407ad2antlr 725 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑀 ∈ LMod)
411, 4, 2, 6lmodvscl 20185 . . . . 5 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€))
4240, 20, 21, 41syl3anc 1371 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€))
43 fvco2 6897 . . . 4 ((𝐺 Fn (Baseβ€˜π‘€) ∧ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€)) β†’ ((𝐹 ∘ 𝐺)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (πΉβ€˜(πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦))))
4439, 42, 43syl2an2r 683 . . 3 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((𝐹 ∘ 𝐺)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (πΉβ€˜(πΊβ€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦))))
45 fvco2 6897 . . . . 5 ((𝐺 Fn (Baseβ€˜π‘€) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ ((𝐹 ∘ 𝐺)β€˜π‘¦) = (πΉβ€˜(πΊβ€˜π‘¦)))
4639, 21, 45syl2an2r 683 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((𝐹 ∘ 𝐺)β€˜π‘¦) = (πΉβ€˜(πΊβ€˜π‘¦)))
4746oveq2d 7323 . . 3 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (π‘₯( ·𝑠 β€˜π‘‚)((𝐹 ∘ 𝐺)β€˜π‘¦)) = (π‘₯( ·𝑠 β€˜π‘‚)(πΉβ€˜(πΊβ€˜π‘¦))))
4838, 44, 473eqtr4d 2786 . 2 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ ((𝐹 ∘ 𝐺)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (π‘₯( ·𝑠 β€˜π‘‚)((𝐹 ∘ 𝐺)β€˜π‘¦)))
491, 2, 3, 4, 5, 6, 8, 10, 14, 18, 48islmhmd 20346 1 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘ 𝐺) ∈ (𝑀 LMHom 𝑂))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104   ∘ ccom 5604   Fn wfn 6453  βŸΆwf 6454  β€˜cfv 6458  (class class class)co 7307  Basecbs 16957  Scalarcsca 17010   ·𝑠 cvsca 17011   GrpHom cghm 18876  LModclmod 20168   LMHom clmhm 20326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-map 8648  df-0g 17197  df-mgm 18371  df-sgrp 18420  df-mnd 18431  df-mhm 18475  df-grp 18625  df-ghm 18877  df-lmod 20170  df-lmhm 20329
This theorem is referenced by:  lmimco  21096  nmhmco  23965  mendring  41055
  Copyright terms: Public domain W3C validator