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Theorem lmhmplusg 21060
Description: The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
lmhmplusg.p + = (+g𝑁)
Assertion
Ref Expression
lmhmplusg ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹f + 𝐺) ∈ (𝑀 LMHom 𝑁))

Proof of Theorem lmhmplusg
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 ( ·𝑠𝑁) = ( ·𝑠𝑁)
4 eqid 2734 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 eqid 2734 . 2 (Scalar‘𝑁) = (Scalar‘𝑁)
6 eqid 2734 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 21049 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
87adantr 480 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 21048 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
109adantr 480 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
114, 5lmhmsca 21046 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
1211adantr 480 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑁) = (Scalar‘𝑀))
13 lmodabl 20923 . . . 4 (𝑁 ∈ LMod → 𝑁 ∈ Abel)
1410, 13syl 17 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ Abel)
15 lmghm 21047 . . . 4 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
1615adantr 480 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
17 lmghm 21047 . . . 4 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
1817adantl 481 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
19 lmhmplusg.p . . . 4 + = (+g𝑁)
2019ghmplusg 19878 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
2114, 16, 18, 20syl3anc 1370 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
22 simpll 767 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 ∈ (𝑀 LMHom 𝑁))
23 simprl 771 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
24 simprr 773 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
254, 6, 1, 2, 3lmhmlin 21051 . . . . . 6 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐹𝑦)))
2622, 23, 24, 25syl3anc 1370 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐹𝑦)))
27 simplr 769 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 ∈ (𝑀 LMHom 𝑁))
284, 6, 1, 2, 3lmhmlin 21051 . . . . . 6 ((𝐺 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐺𝑦)))
2927, 23, 24, 28syl3anc 1370 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐺𝑦)))
3026, 29oveq12d 7448 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹‘(𝑥( ·𝑠𝑀)𝑦)) + (𝐺‘(𝑥( ·𝑠𝑀)𝑦))) = ((𝑥( ·𝑠𝑁)(𝐹𝑦)) + (𝑥( ·𝑠𝑁)(𝐺𝑦))))
319ad2antrr 726 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ LMod)
3211fveq2d 6910 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
3332ad2antrr 726 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
3423, 33eleqtrrd 2841 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
35 eqid 2734 . . . . . . . 8 (Base‘𝑁) = (Base‘𝑁)
361, 35lmhmf 21050 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
3736ad2antrr 726 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
3837, 24ffvelcdmd 7104 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑦) ∈ (Base‘𝑁))
391, 35lmhmf 21050 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
4039ad2antlr 727 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
4140, 24ffvelcdmd 7104 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑦) ∈ (Base‘𝑁))
42 eqid 2734 . . . . . 6 (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
4335, 19, 5, 3, 42lmodvsdi 20899 . . . . 5 ((𝑁 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑁)) ∧ (𝐹𝑦) ∈ (Base‘𝑁) ∧ (𝐺𝑦) ∈ (Base‘𝑁))) → (𝑥( ·𝑠𝑁)((𝐹𝑦) + (𝐺𝑦))) = ((𝑥( ·𝑠𝑁)(𝐹𝑦)) + (𝑥( ·𝑠𝑁)(𝐺𝑦))))
4431, 34, 38, 41, 43syl13anc 1371 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠𝑁)((𝐹𝑦) + (𝐺𝑦))) = ((𝑥( ·𝑠𝑁)(𝐹𝑦)) + (𝑥( ·𝑠𝑁)(𝐺𝑦))))
4530, 44eqtr4d 2777 . . 3 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹‘(𝑥( ·𝑠𝑀)𝑦)) + (𝐺‘(𝑥( ·𝑠𝑀)𝑦))) = (𝑥( ·𝑠𝑁)((𝐹𝑦) + (𝐺𝑦))))
4637ffnd 6737 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
4740ffnd 6737 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
48 fvexd 6921 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
497ad2antrr 726 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
501, 4, 2, 6lmodvscl 20892 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))
5149, 23, 24, 50syl3anc 1370 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))
52 fnfvof 7713 . . . 4 (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘(𝑥( ·𝑠𝑀)𝑦)) = ((𝐹‘(𝑥( ·𝑠𝑀)𝑦)) + (𝐺‘(𝑥( ·𝑠𝑀)𝑦))))
5346, 47, 48, 51, 52syl22anc 839 . . 3 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘(𝑥( ·𝑠𝑀)𝑦)) = ((𝐹‘(𝑥( ·𝑠𝑀)𝑦)) + (𝐺‘(𝑥( ·𝑠𝑀)𝑦))))
54 fnfvof 7713 . . . . 5 (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5546, 47, 48, 24, 54syl22anc 839 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5655oveq2d 7446 . . 3 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠𝑁)((𝐹f + 𝐺)‘𝑦)) = (𝑥( ·𝑠𝑁)((𝐹𝑦) + (𝐺𝑦))))
5745, 53, 563eqtr4d 2784 . 2 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹f + 𝐺)‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)((𝐹f + 𝐺)‘𝑦)))
581, 2, 3, 4, 5, 6, 8, 10, 12, 21, 57islmhmd 21055 1 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹f + 𝐺) ∈ (𝑀 LMHom 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694  Basecbs 17244  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301   GrpHom cghm 19242  Abelcabl 19813  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-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3044  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-pss 3982  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-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  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-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-plusg 17310  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-ghm 19243  df-cmn 19814  df-abl 19815  df-mgp 20152  df-ur 20199  df-ring 20252  df-lmod 20876  df-lmhm 21038
This theorem is referenced by:  nmhmplusg  24793  mendring  43176
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