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Theorem lmhmplusg 19256
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 2770 . 2 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2770 . 2 ( ·𝑠𝑀) = ( ·𝑠𝑀)
3 eqid 2770 . 2 ( ·𝑠𝑁) = ( ·𝑠𝑁)
4 eqid 2770 . 2 (Scalar‘𝑀) = (Scalar‘𝑀)
5 eqid 2770 . 2 (Scalar‘𝑁) = (Scalar‘𝑁)
6 eqid 2770 . 2 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7 lmhmlmod1 19245 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
87adantr 466 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9 lmhmlmod2 19244 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
109adantr 466 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
114, 5lmhmsca 19242 . . 3 (𝐹 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
1211adantr 466 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑁) = (Scalar‘𝑀))
13 lmodabl 19119 . . . 4 (𝑁 ∈ LMod → 𝑁 ∈ Abel)
1410, 13syl 17 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ Abel)
15 lmghm 19243 . . . 4 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
1615adantr 466 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
17 lmghm 19243 . . . 4 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
1817adantl 467 . . 3 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
19 lmhmplusg.p . . . 4 + = (+g𝑁)
2019ghmplusg 18455 . . 3 ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹𝑓 + 𝐺) ∈ (𝑀 GrpHom 𝑁))
2114, 16, 18, 20syl3anc 1475 . 2 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝑓 + 𝐺) ∈ (𝑀 GrpHom 𝑁))
22 simpll 742 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 ∈ (𝑀 LMHom 𝑁))
23 simprl 746 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
24 simprr 748 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
254, 6, 1, 2, 3lmhmlin 19247 . . . . . 6 ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐹𝑦)))
2622, 23, 24, 25syl3anc 1475 . . . . 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 744 . . . . . 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 19247 . . . . . 6 ((𝐺 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥( ·𝑠𝑀)𝑦)) = (𝑥( ·𝑠𝑁)(𝐺𝑦)))
2927, 23, 24, 28syl3anc 1475 . . . . 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 6810 . . . 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 697 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ LMod)
3211fveq2d 6336 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
3332ad2antrr 697 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
3423, 33eleqtrrd 2852 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
35 eqid 2770 . . . . . . . 8 (Base‘𝑁) = (Base‘𝑁)
361, 35lmhmf 19246 . . . . . . 7 (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
3736ad2antrr 697 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
3837, 24ffvelrnd 6503 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹𝑦) ∈ (Base‘𝑁))
391, 35lmhmf 19246 . . . . . . 7 (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
4039ad2antlr 698 . . . . . 6 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
4140, 24ffvelrnd 6503 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺𝑦) ∈ (Base‘𝑁))
42 eqid 2770 . . . . . 6 (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
4335, 19, 5, 3, 42lmodvsdi 19095 . . . . 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 1477 . . . 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 2807 . . 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))))
46 ffn 6185 . . . . 5 (𝐹:(Base‘𝑀)⟶(Base‘𝑁) → 𝐹 Fn (Base‘𝑀))
4737, 46syl 17 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
48 ffn 6185 . . . . 5 (𝐺:(Base‘𝑀)⟶(Base‘𝑁) → 𝐺 Fn (Base‘𝑀))
4940, 48syl 17 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
50 fvexd 6344 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
517ad2antrr 697 . . . . 5 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
521, 4, 2, 6lmodvscl 19089 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))
5351, 23, 24, 52syl3anc 1475 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠𝑀)𝑦) ∈ (Base‘𝑀))
54 fnfvof 7057 . . . 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))))
5547, 49, 50, 53, 54syl22anc 1476 . . 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))))
56 fnfvof 7057 . . . . 5 (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5747, 49, 50, 24, 56syl22anc 1476 . . . 4 (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹𝑓 + 𝐺)‘𝑦) = ((𝐹𝑦) + (𝐺𝑦)))
5857oveq2d 6808 . . 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))))
5945, 55, 583eqtr4d 2814 . 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)))
601, 2, 3, 4, 5, 6, 8, 10, 12, 21, 59islmhmd 19251 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 382   = wceq 1630  wcel 2144  Vcvv 3349   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6792  𝑓 cof 7041  Basecbs 16063  +gcplusg 16148  Scalarcsca 16151   ·𝑠 cvsca 16152   GrpHom cghm 17864  Abelcabl 18400  LModclmod 19072   LMHom clmhm 19231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-plusg 16161  df-0g 16309  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-minusg 17633  df-ghm 17865  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-lmod 19074  df-lmhm 19234
This theorem is referenced by:  nmhmplusg  22780  mendring  38281
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