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Theorem lmhmco 19091
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 2651 . 2 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2651 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 eqid 2651 . 2 ( ·𝑠𝑂) = ( ·𝑠𝑂)
4 eqid 2651 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 eqid 2651 . 2 (Scalar‘𝑂) = (Scalar‘𝑂)
6 eqid 2651 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 19081 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
87adantl 481 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 19080 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝑂 ∈ LMod)
109adantr 480 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑂 ∈ LMod)
11 eqid 2651 . . . 4 (Scalar‘𝑁) = (Scalar‘𝑁)
1211, 5lmhmsca 19078 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) → (Scalar‘𝑂) = (Scalar‘𝑁))
134, 11lmhmsca 19078 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
1412, 13sylan9eq 2705 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑂) = (Scalar‘𝑀))
15 lmghm 19079 . . 3 (𝐹 ∈ (𝑁 LMHom 𝑂) → 𝐹 ∈ (𝑁 GrpHom 𝑂))
16 lmghm 19079 . . 3 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
17 ghmco 17727 . . 3 ((𝐹 ∈ (𝑁 GrpHom 𝑂) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝐺) ∈ (𝑀 GrpHom 𝑂))
1815, 16, 17syl2an 493 . 2 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝐺) ∈ (𝑀 GrpHom 𝑂))
19 simplr 807 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 ∈ (𝑀 LMHom 𝑁))
20 simprl 809 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
21 simprr 811 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
22 eqid 2651 . . . . . . 7 ( ·𝑠𝑁) = ( ·𝑠𝑁)
234, 6, 1, 2, 22lmhmlin 19083 . . . . . 6 ((𝐺 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐺𝑦)))
2419, 20, 21, 23syl3anc 1366 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐺𝑦)))
2524fveq2d 6233 . . . 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 805 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 ∈ (𝑁 LMHom 𝑂))
2713fveq2d 6233 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
2827ad2antlr 763 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
2920, 28eleqtrrd 2733 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
30 eqid 2651 . . . . . . . . 9 (Base‘𝑁) = (Base‘𝑁)
311, 30lmhmf 19082 . . . . . . . 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))
3332ffvelrnda 6399 . . . . . 6 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺𝑦) ∈ (Base‘𝑁))
3433adantrl 752 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑦) ∈ (Base‘𝑁))
35 eqid 2651 . . . . . 6 (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
3611, 35, 30, 22, 3lmhmlin 19083 . . . . 5 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑁)) ∧ (𝐺𝑦) ∈ (Base‘𝑁)) → (𝐹‘(𝑥( ·𝑠𝑁)(𝐺𝑦))) = (𝑥( ·𝑠𝑂)(𝐹‘(𝐺𝑦))))
3726, 29, 34, 36syl3anc 1366 . . . 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 2685 . . 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))))
39 ffn 6083 . . . . . 6 (𝐺:(Base‘𝑀)⟶(Base‘𝑁) → 𝐺 Fn (Base‘𝑀))
4032, 39syl 17 . . . . 5 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 Fn (Base‘𝑀))
4140adantr 480 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
427ad2antlr 763 . . . . 5 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
431, 4, 2, 6lmodvscl 18928 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))
4442, 20, 21, 43syl3anc 1366 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))
45 fvco2 6312 . . . 4 ((𝐺 Fn (Base‘𝑀) ∧ (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀)) → ((𝐹𝐺)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝐹‘(𝐺‘(𝑥( ·𝑠𝑀)𝑦))))
4641, 44, 45syl2anc 694 . . 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))))
47 fvco2 6312 . . . . 5 ((𝐺 Fn (Base‘𝑀) ∧ 𝑦 ∈ (Base‘𝑀)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
4841, 21, 47syl2anc 694 . . . 4 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
4948oveq2d 6706 . . 3 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠𝑂)((𝐹𝐺)‘𝑦)) = (𝑥( ·𝑠𝑂)(𝐹‘(𝐺𝑦))))
5038, 46, 493eqtr4d 2695 . 2 (((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝐺)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑂)((𝐹𝐺)‘𝑦)))
511, 2, 3, 4, 5, 6, 8, 10, 14, 18, 50islmhmd 19087 1 ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝐺) ∈ (𝑀 LMHom 𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992   GrpHom cghm 17704  LModclmod 18911   LMHom clmhm 19067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-grp 17472  df-ghm 17705  df-lmod 18913  df-lmhm 19070
This theorem is referenced by:  lmimco  20231  nmhmco  22607  mendring  38079
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