Theorem lmhmplusg 20951

Description: The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypothesis

Ref	Expression
lmhmplusg.p	⊢ + = (+g‘𝑁)

Assertion

Ref	Expression
lmhmplusg	⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁))

Proof of Theorem lmhmplusg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2729	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2729	. 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁)
4		eqid 2729	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		eqid 2729	. 2 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁)
6		eqid 2729	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 20940	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 20939	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
10	9	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
11	4, 5	lmhmsca 20937	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
12	11	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑁) = (Scalar‘𝑀))
13		lmodabl 20815	. . . 4 ⊢ (𝑁 ∈ LMod → 𝑁 ∈ Abel)
14	10, 13	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ Abel)
15		lmghm 20938	. . . 4 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
16	15	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
17		lmghm 20938	. . . 4 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
18	17	adantl 481	. . 3 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
19		lmhmplusg.p	. . . 4 ⊢ + = (+g‘𝑁)
20	19	ghmplusg 19776	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
21	14, 16, 18, 20	syl3anc 1373	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
22		simpll 766	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 ∈ (𝑀 LMHom 𝑁))
23		simprl 770	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑀)))
24		simprr 772	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑦 ∈ (Base‘𝑀))
25	4, 6, 1, 2, 3	lmhmlin 20942	. . . . . 6 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)(𝐹‘𝑦)))
26	22, 23, 24, 25	syl3anc 1373	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)(𝐹‘𝑦)))
27		simplr 768	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 ∈ (𝑀 LMHom 𝑁))
28	4, 6, 1, 2, 3	lmhmlin 20942	. . . . . 6 ⊢ ((𝐺 ∈ (𝑀 LMHom 𝑁) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦)))
29	27, 23, 24, 28	syl3anc 1373	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦)))
30	26, 29	oveq12d 7405	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) + (𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦))) = ((𝑥( ·𝑠 ‘𝑁)(𝐹‘𝑦)) + (𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦))))
31	9	ad2antrr 726	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ LMod)
32	11	fveq2d 6862	. . . . . . 7 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
33	32	ad2antrr 726	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
34	23, 33	eleqtrrd 2831	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
35		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝑁) = (Base‘𝑁)
36	1, 35	lmhmf 20941	. . . . . . 7 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
37	36	ad2antrr 726	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
38	37, 24	ffvelcdmd 7057	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑦) ∈ (Base‘𝑁))
39	1, 35	lmhmf 20941	. . . . . . 7 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
40	39	ad2antlr 727	. . . . . 6 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
41	40, 24	ffvelcdmd 7057	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑦) ∈ (Base‘𝑁))
42		eqid 2729	. . . . . 6 ⊢ (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
43	35, 19, 5, 3, 42	lmodvsdi 20791	. . . . 5 ⊢ ((𝑁 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑁)) ∧ (𝐹‘𝑦) ∈ (Base‘𝑁) ∧ (𝐺‘𝑦) ∈ (Base‘𝑁))) → (𝑥( ·𝑠 ‘𝑁)((𝐹‘𝑦) + (𝐺‘𝑦))) = ((𝑥( ·𝑠 ‘𝑁)(𝐹‘𝑦)) + (𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦))))
44	31, 34, 38, 41, 43	syl13anc 1374	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑁)((𝐹‘𝑦) + (𝐺‘𝑦))) = ((𝑥( ·𝑠 ‘𝑁)(𝐹‘𝑦)) + (𝑥( ·𝑠 ‘𝑁)(𝐺‘𝑦))))
45	30, 44	eqtr4d 2767	. . 3 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) + (𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦))) = (𝑥( ·𝑠 ‘𝑁)((𝐹‘𝑦) + (𝐺‘𝑦))))
46	37	ffnd 6689	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
47	40	ffnd 6689	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
48		fvexd 6873	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
49	7	ad2antrr 726	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
50	1, 4, 2, 6	lmodvscl 20784	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
51	49, 23, 24, 50	syl3anc 1373	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
52		fnfvof 7670	. . . 4 ⊢ (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = ((𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) + (𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦))))
53	46, 47, 48, 51, 52	syl22anc 838	. . 3 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = ((𝐹‘(𝑥( ·𝑠 ‘𝑀)𝑦)) + (𝐺‘(𝑥( ·𝑠 ‘𝑀)𝑦))))
54		fnfvof 7670	. . . . 5 ⊢ (((𝐹 Fn (Base‘𝑀) ∧ 𝐺 Fn (Base‘𝑀)) ∧ ((Base‘𝑀) ∈ V ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘𝑦) = ((𝐹‘𝑦) + (𝐺‘𝑦)))
55	46, 47, 48, 24, 54	syl22anc 838	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘𝑦) = ((𝐹‘𝑦) + (𝐺‘𝑦)))
56	55	oveq2d 7403	. . 3 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑁)((𝐹 ∘f + 𝐺)‘𝑦)) = (𝑥( ·𝑠 ‘𝑁)((𝐹‘𝑦) + (𝐺‘𝑦))))
57	45, 53, 56	3eqtr4d 2774	. 2 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → ((𝐹 ∘f + 𝐺)‘(𝑥( ·𝑠 ‘𝑀)𝑦)) = (𝑥( ·𝑠 ‘𝑁)((𝐹 ∘f + 𝐺)‘𝑦)))
58	1, 2, 3, 4, 5, 6, 8, 10, 12, 21, 57	islmhmd 20946	1 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651  Basecbs 17179  +_gcplusg 17220  Scalarcsca 17223   ·_𝑠 cvsca 17224   GrpHom cghm 19144  Abelcabl 19711  LModclmod 20766   LMHom clmhm 20926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-ghm 19145  df-cmn 19712  df-abl 19713  df-mgp 20050  df-ur 20091  df-ring 20144  df-lmod 20768  df-lmhm 20929

This theorem is referenced by:  nmhmplusg  24645  mendring  43177