Theorem lmhmplusg 21031

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 2737	. 2 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2737	. 2 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
3		eqid 2737	. 2 ⊢ ( ·𝑠 ‘𝑁) = ( ·𝑠 ‘𝑁)
4		eqid 2737	. 2 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
5		eqid 2737	. 2 ⊢ (Scalar‘𝑁) = (Scalar‘𝑁)
6		eqid 2737	. 2 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
7		lmhmlmod1 21020	. . 3 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝑀 ∈ LMod)
8	7	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑀 ∈ LMod)
9		lmhmlmod2 21019	. . 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 21017	. . 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 20895	. . . 4 ⊢ (𝑁 ∈ LMod → 𝑁 ∈ Abel)
14	10, 13	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → 𝑁 ∈ Abel)
15		lmghm 21018	. . . 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 21018	. . . 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 19812	. . 3 ⊢ ((𝑁 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝐺 ∈ (𝑀 GrpHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 GrpHom 𝑁))
21	14, 16, 18, 20	syl3anc 1374	. 2 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘f + 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))
25	4, 6, 1, 2, 3	lmhmlin 21022	. . . . . 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 1374	. . . . 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 769	. . . . . 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 21022	. . . . . 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 1374	. . . . 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 7378	. . . 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 727	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑁 ∈ LMod)
32	11	fveq2d 6838	. . . . . . 7 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑀)))
33	32	ad2antrr 727	. . . . . 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 2840	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑥 ∈ (Base‘(Scalar‘𝑁)))
35		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑁) = (Base‘𝑁)
36	1, 35	lmhmf 21021	. . . . . . 7 ⊢ (𝐹 ∈ (𝑀 LMHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
37	36	ad2antrr 727	. . . . . 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 7031	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐹‘𝑦) ∈ (Base‘𝑁))
39	1, 35	lmhmf 21021	. . . . . . 7 ⊢ (𝐺 ∈ (𝑀 LMHom 𝑁) → 𝐺:(Base‘𝑀)⟶(Base‘𝑁))
40	39	ad2antlr 728	. . . . . 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 7031	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝐺‘𝑦) ∈ (Base‘𝑁))
42		eqid 2737	. . . . . 6 ⊢ (Base‘(Scalar‘𝑁)) = (Base‘(Scalar‘𝑁))
43	35, 19, 5, 3, 42	lmodvsdi 20871	. . . . 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 1375	. . . 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 2775	. . 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 6663	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐹 Fn (Base‘𝑀))
47	40	ffnd 6663	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝐺 Fn (Base‘𝑀))
48		fvexd 6849	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (Base‘𝑀) ∈ V)
49	7	ad2antrr 727	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → 𝑀 ∈ LMod)
50	1, 4, 2, 6	lmodvscl 20864	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
51	49, 23, 24, 50	syl3anc 1374	. . . 4 ⊢ (((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑀)) ∧ 𝑦 ∈ (Base‘𝑀))) → (𝑥( ·𝑠 ‘𝑀)𝑦) ∈ (Base‘𝑀))
52		fnfvof 7641	. . . 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 839	. . 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 7641	. . . . 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 839	. . . 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 7376	. . 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 2782	. 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 21026	1 ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ∘_f cof 7622  Basecbs 17170  +_gcplusg 17211  Scalarcsca 17214   ·_𝑠 cvsca 17215   GrpHom cghm 19178  Abelcabl 19747  LModclmod 20846   LMHom clmhm 21006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-plusg 17224  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-ghm 19179  df-cmn 19748  df-abl 19749  df-mgp 20113  df-ur 20154  df-ring 20207  df-lmod 20848  df-lmhm 21009

This theorem is referenced by:  nmhmplusg  24732  mendring  43634