Theorem lmhmplusg 20632

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 2733	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2733	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2733	. 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁)
4		eqid 2733	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		eqid 2733	. 2 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁)
6		eqid 2733	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 20621	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	adantr 482	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 20620	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑁 ∈ LMod)
10	9	adantr 482	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ LMod)
11	4, 5	lmhmsca 20618	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → (Scalar‘𝑁) = (Scalar‘𝑀))
12	11	adantr 482	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (Scalar‘𝑁) = (Scalar‘𝑀))
13		lmodabl 20496	. . . 4 ⊢ (𝑁 ∈ LMod → 𝑁 ∈ Abel)
14	10, 13	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ Abel)
15		lmghm 20619	. . . 4 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
16	15	adantr 482	. . 3 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
17		lmghm 20619	. . . 4 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
18	17	adantl 483	. . 3 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝐺 ∈ (𝑀 GrpHom 𝑁))
19		lmhmplusg.p	. . . 4 ⊢ + = (+g‘𝑁)
20	19	ghmplusg 19697	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
21	14, 16, 18, 20	syl3anc 1372	. 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 20623	. . . . . 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 1372	. . . . 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 20623	. . . . . 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 1372	. . . . 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 7414	. . . 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 725	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ LMod)
32	11	fveq2d 6885	. . . . . . 7 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
33	32	ad2antrr 725	. . . . . 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 2733	. . . . . . . 8 ⊢ (Base‘𝑁) = (Base‘𝑁)
36	1, 35	lmhmf 20622	. . . . . . 7 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
37	36	ad2antrr 725	. . . . . 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 20622	. . . . . . 7 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
40	39	ad2antlr 726	. . . . . 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 2733	. . . . . 6 ⊢ (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
43	35, 19, 5, 3, 42	lmodvsdi 20472	. . . . 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 1373	. . . 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 2776	. . 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 6708	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
47	40	ffnd 6708	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
48		fvexd 6896	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
49	7	ad2antrr 725	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
50	1, 4, 2, 6	lmodvscl 20466	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
51	49, 23, 24, 50	syl3anc 1372	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
52		fnfvof 7674	. . . 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 7674	. . . . 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 7412	. . 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 2783	. 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 20627	1 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   Fn wfn 6530  ⟶wf 6531  ‘cfv 6535  (class class class)co 7396   ∘_f cof 7655  Basecbs 17131  +_gcplusg 17184  Scalarcsca 17187   ·_𝑠 cvsca 17188   GrpHom cghm 19074  Abelcabl 19633  LModclmod 20448   LMHom clmhm 20607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7657  df-om 7843  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-er 8691  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-2 12262  df-sets 17084  df-slot 17102  df-ndx 17114  df-base 17132  df-plusg 17197  df-0g 17374  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-grp 18809  df-minusg 18810  df-ghm 19075  df-cmn 19634  df-abl 19635  df-mgp 19971  df-ur 19988  df-ring 20040  df-lmod 20450  df-lmhm 20610

This theorem is referenced by:  nmhmplusg  24243  mendring  41805