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Theorem lmhmco 21059
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)) → (𝐹𝐺) ∈ (𝑀 LMHom 𝑂))

Proof of Theorem lmhmco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . 2 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2734 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 eqid 2734 . 2 ( ·𝑠𝑂) = ( ·𝑠𝑂)
4 eqid 2734 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 eqid 2734 . 2 (Scalar‘𝑂) = (Scalar‘𝑂)
6 eqid 2734 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 21049 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
87adantl 481 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 21048 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝑂 ∈ LMod)
109adantr 480 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑂 ∈ LMod)
11 eqid 2734 . . . 4 (Scalar‘𝑁) = (Scalar‘𝑁)
1211, 5lmhmsca 21046 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) → (Scalar‘𝑂) = (Scalar‘𝑁))
134, 11lmhmsca 21046 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
1412, 13sylan9eq 2794 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑂) = (Scalar‘𝑀))
15 lmghm 21047 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝐹 ∈ (𝑁 GrpHom 𝑂))
16 lmghm 21047 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
17 ghmco 19266 . . 3 ((𝐹 ∈ (𝑁 GrpHom 𝑂) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝐺) ∈ (𝑀 GrpHom 𝑂))
1815, 16, 17syl2an 596 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝐺) ∈ (𝑀 GrpHom 𝑂))
19 simplr 769 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 ∈ (𝑀 LMHom 𝑁))
20 simprl 771 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
21 simprr 773 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
22 eqid 2734 . . . . . . 7 ( ·𝑠𝑁) = ( ·𝑠𝑁)
234, 6, 1, 2, 22lmhmlin 21051 . . . . . 6 ((𝐺 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐺𝑦)))
2419, 20, 21, 23syl3anc 1370 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐺𝑦)))
2524fveq2d 6910 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝐺‘(𝑥( ·𝑠𝑀)𝑦))) = (𝐹‘(𝑥( ·𝑠𝑁)(𝐺𝑦))))
26 simpll 767 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 ∈ (𝑁 LMHom 𝑂))
2713fveq2d 6910 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
2827ad2antlr 727 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
2920, 28eleqtrrd 2841 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
30 eqid 2734 . . . . . . . . 9 (Base‘𝑁) = (Base‘𝑁)
311, 30lmhmf 21050 . . . . . . . 8 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
3231adantl 481 . . . . . . 7 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
3332ffvelcdmda 7103 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺𝑦) ∈ (Base‘𝑁))
3433adantrl 716 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑦) ∈ (Base‘𝑁))
35 eqid 2734 . . . . . 6 (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
3611, 35, 30, 22, 3lmhmlin 21051 . . . . 5 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑁)) ∧ (𝐺𝑦) ∈ (Base‘𝑁)) → (𝐹‘(𝑥( ·𝑠𝑁)(𝐺𝑦))) = (𝑥( ·𝑠𝑂)(𝐹‘(𝐺𝑦))))
3726, 29, 34, 36syl3anc 1370 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥( ·𝑠𝑁)(𝐺𝑦))) = (𝑥( ·𝑠𝑂)(𝐹‘(𝐺𝑦))))
3825, 37eqtrd 2774 . . 3 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝐺‘(𝑥( ·𝑠𝑀)𝑦))) = (𝑥( ·𝑠𝑂)(𝐹‘(𝐺𝑦))))
3932ffnd 6737 . . . 4 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
407ad2antlr 727 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
411, 4, 2, 6lmodvscl 20892 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))
4240, 20, 21, 41syl3anc 1370 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))
43 fvco2 7005 . . . 4 ((𝐺 Fn (Base‘𝑀) ∧ (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀)) → ((𝐹𝐺)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐹‘(𝐺‘(𝑥( ·𝑠𝑀)𝑦))))
4439, 42, 43syl2an2r 685 . . 3 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝐺)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐹‘(𝐺‘(𝑥( ·𝑠𝑀)𝑦))))
45 fvco2 7005 . . . . 5 ((𝐺 Fn (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
4639, 21, 45syl2an2r 685 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
4746oveq2d 7446 . . 3 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠𝑂)((𝐹𝐺)‘𝑦)) = (𝑥( ·𝑠𝑂)(𝐹‘(𝐺𝑦))))
4838, 44, 473eqtr4d 2784 . 2 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝐺)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑂)((𝐹𝐺)‘𝑦)))
491, 2, 3, 4, 5, 6, 8, 10, 14, 18, 48islmhmd 21055 1 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝐺) ∈ (𝑀 LMHom 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  Basecbs 17244  Scalarcsca 17300   ·𝑠 cvsca 17301   GrpHom cghm 19242  LModclmod 20874   LMHom clmhm 21035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-grp 18966  df-ghm 19243  df-lmod 20876  df-lmhm 21038
This theorem is referenced by:  lmimco  21881  nmhmco  24792  mendring  43176
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