Theorem lmhmplusg 21002

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 2735	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2735	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2735	. 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁)
4		eqid 2735	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		eqid 2735	. 2 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁)
6		eqid 2735	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 20991	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 20990	. . 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 20988	. . 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 20866	. . . 4 ⊢ (𝑁 ∈ LMod → 𝑁 ∈ Abel)
14	10, 13	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ Abel)
15		lmghm 20989	. . . 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 20989	. . . 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 19827	. . 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 20993	. . . . . 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 20993	. . . . . 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 7423	. . . 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 6880	. . . . . . 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 2837	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
35		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝑁) = (Base‘𝑁)
36	1, 35	lmhmf 20992	. . . . . . 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 7075	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑦) ∈ (Base‘𝑁))
39	1, 35	lmhmf 20992	. . . . . . 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 7075	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑦) ∈ (Base‘𝑁))
42		eqid 2735	. . . . . 6 ⊢ (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
43	35, 19, 5, 3, 42	lmodvsdi 20842	. . . . 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 2773	. . 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 6707	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
47	40	ffnd 6707	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
48		fvexd 6891	. . . 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 20835	. . . . 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 7688	. . . 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 7688	. . . . 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 7421	. . 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 2780	. 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 20997	1 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405   ∘_f cof 7669  Basecbs 17228  +_gcplusg 17271  Scalarcsca 17274   ·_𝑠 cvsca 17275   GrpHom cghm 19195  Abelcabl 19762  LModclmod 20817   LMHom clmhm 20977

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-plusg 17284  df-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-grp 18919  df-minusg 18920  df-ghm 19196  df-cmn 19763  df-abl 19764  df-mgp 20101  df-ur 20142  df-ring 20195  df-lmod 20819  df-lmhm 20980

This theorem is referenced by:  nmhmplusg  24696  mendring  43212