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Theorem lmhmco 20654
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 2733 . 2 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2733 . 2 ( ·𝑠 β€˜π‘€) = ( ·𝑠 β€˜π‘€)
3 eqid 2733 . 2 ( ·𝑠 β€˜π‘‚) = ( ·𝑠 β€˜π‘‚)
4 eqid 2733 . 2 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
5 eqid 2733 . 2 (Scalarβ€˜π‘‚) = (Scalarβ€˜π‘‚)
6 eqid 2733 . 2 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
7 lmhmlmod1 20644 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝑀 ∈ LMod)
87adantl 483 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑀 ∈ LMod)
9 lmhmlmod2 20643 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) β†’ 𝑂 ∈ LMod)
109adantr 482 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝑂 ∈ LMod)
11 eqid 2733 . . . 4 (Scalarβ€˜π‘) = (Scalarβ€˜π‘)
1211, 5lmhmsca 20641 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) β†’ (Scalarβ€˜π‘‚) = (Scalarβ€˜π‘))
134, 11lmhmsca 20641 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ (Scalarβ€˜π‘) = (Scalarβ€˜π‘€))
1412, 13sylan9eq 2793 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (Scalarβ€˜π‘‚) = (Scalarβ€˜π‘€))
15 lmghm 20642 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) β†’ 𝐹 ∈ (𝑁 GrpHom 𝑂))
16 lmghm 20642 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ 𝐺 ∈ (𝑀 GrpHom 𝑁))
17 ghmco 19112 . . 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 768 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐺 ∈ (𝑀 LMHom 𝑁))
20 simprl 770 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)))
21 simprr 772 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑦 ∈ (Baseβ€˜π‘€))
22 eqid 2733 . . . . . . 7 ( ·𝑠 β€˜π‘) = ( ·𝑠 β€˜π‘)
234, 6, 1, 2, 22lmhmlin 20646 . . . . . 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 1372 . . . . 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 6896 . . . 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 766 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝐹 ∈ (𝑁 LMHom 𝑂))
2713fveq2d 6896 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) β†’ (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘€)))
2827ad2antlr 726 . . . . . 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 2837 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘)))
30 eqid 2733 . . . . . . . . 9 (Baseβ€˜π‘) = (Baseβ€˜π‘)
311, 30lmhmf 20645 . . . . . . . 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 7087 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘))
3433adantrl 715 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ (πΊβ€˜π‘¦) ∈ (Baseβ€˜π‘))
35 eqid 2733 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘)) = (Baseβ€˜(Scalarβ€˜π‘))
3611, 35, 30, 22, 3lmhmlin 20646 . . . . 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 1372 . . . 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 2773 . . 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 6719 . . . 4 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ 𝐺 Fn (Baseβ€˜π‘€))
407ad2antlr 726 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€))) β†’ 𝑀 ∈ LMod)
411, 4, 2, 6lmodvscl 20489 . . . . 5 ((𝑀 ∈ LMod ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘€)) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ (π‘₯( ·𝑠 β€˜π‘€)𝑦) ∈ (Baseβ€˜π‘€))
4240, 20, 21, 41syl3anc 1372 . . . 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 6989 . . . 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 684 . . 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 6989 . . . . 5 ((𝐺 Fn (Baseβ€˜π‘€) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) β†’ ((𝐹 ∘ 𝐺)β€˜π‘¦) = (πΉβ€˜(πΊβ€˜π‘¦)))
4639, 21, 45syl2an2r 684 . . . 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 7425 . . 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 2783 . 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 20650 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 1542   ∈ wcel 2107   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201   GrpHom cghm 19089  LModclmod 20471   LMHom clmhm 20630
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-rmo 3377  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-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-grp 18822  df-ghm 19090  df-lmod 20473  df-lmhm 20633
This theorem is referenced by:  lmimco  21399  nmhmco  24273  mendring  41934
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