Theorem lincsum 48554

Description: The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 28-Jul-2019.)

Hypotheses

Ref	Expression
lincsum.p	⊢ + = (+g‘𝑀)
lincsum.x	⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉)
lincsum.y	⊢ 𝑌 = (𝐵( linC ‘𝑀)𝑉)
lincsum.s	⊢ 𝑆 = (Scalar‘𝑀)
lincsum.r	⊢ 𝑅 = (Base‘𝑆)
lincsum.b	⊢ ✚ = (+g‘𝑆)

Assertion

Ref	Expression
lincsum	⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑋 + 𝑌) = ((𝐴 ∘f ✚ 𝐵)( linC ‘𝑀)𝑉))

Proof of Theorem lincsum

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

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2733	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
3		lincsum.p	. . 3 ⊢ + = (+g‘𝑀)
4		lmodcmn 20845	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
5	4	adantr 480	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ CMnd)
6	5	3ad2ant1 1133	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → 𝑀 ∈ CMnd)
7		simpr 484	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
8	7	3ad2ant1 1133	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
9		simpl 482	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
10	9	3ad2ant1 1133	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → 𝑀 ∈ LMod)
11	10	adantr 480	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod)
12		elmapi 8779	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
13		ffvelcdm 7020	. . . . . . . . 9 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
14	13	ex 412	. . . . . . . 8 ⊢ (𝐴:𝑉⟶𝑅 → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
15	12, 14	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
16	15	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
17	16	3ad2ant2 1134	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
18	17	imp 406	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
19		elelpwi 4559	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
20	19	expcom 413	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
21	20	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
22	21	3ad2ant1 1133	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
23	22	imp 406	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑀))
24		lincsum.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑀)
25		eqid 2733	. . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
26		lincsum.r	. . . . 5 ⊢ 𝑅 = (Base‘𝑆)
27	1, 24, 25, 26	lmodvscl 20813	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴‘𝑥) ∈ 𝑅 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
28	11, 18, 23, 27	syl3anc 1373	. . 3 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) ∧ 𝑥 ∈ 𝑉) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
29		elmapi 8779	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵:𝑉⟶𝑅)
30		ffvelcdm 7020	. . . . . . . . 9 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐵‘𝑥) ∈ 𝑅)
31	30	ex 412	. . . . . . . 8 ⊢ (𝐵:𝑉⟶𝑅 → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
32	29, 31	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
33	32	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
34	33	3ad2ant2 1134	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
35	34	imp 406	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) ∧ 𝑥 ∈ 𝑉) → (𝐵‘𝑥) ∈ 𝑅)
36	1, 24, 25, 26	lmodvscl 20813	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐵‘𝑥) ∈ 𝑅 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
37	11, 35, 23, 36	syl3anc 1373	. . 3 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) ∧ 𝑥 ∈ 𝑉) → ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
38		eqidd 2734	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
39		eqidd 2734	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
40		id 22	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
41		simpl 482	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐴 ∈ (𝑅 ↑m 𝑉))
42		simpl 482	. . . 4 ⊢ ((𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆)) → 𝐴 finSupp (0g‘𝑆))
43	24, 26	scmfsupp 48499	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥)) finSupp (0g‘𝑀))
44	40, 41, 42, 43	syl3an 1160	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥)) finSupp (0g‘𝑀))
45		simpr 484	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐵 ∈ (𝑅 ↑m 𝑉))
46		simpr 484	. . . 4 ⊢ ((𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆)) → 𝐵 finSupp (0g‘𝑆))
47	24, 26	scmfsupp 48499	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 finSupp (0g‘𝑆)) → (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)) finSupp (0g‘𝑀))
48	40, 45, 46, 47	syl3an 1160	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)) finSupp (0g‘𝑀))
49	1, 2, 3, 6, 8, 28, 37, 38, 39, 44, 48	gsummptfsadd 19838	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑀 Σg (𝑥 ∈ 𝑉 ↦ (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) + ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))) = ((𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
50	7	adantr 480	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
51		elmapfn 8795	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 Fn 𝑉)
52	51	ad2antrl 728	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐴 Fn 𝑉)
53		elmapfn 8795	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵 Fn 𝑉)
54	53	ad2antll 729	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵 Fn 𝑉)
55	50, 52, 54	offvalfv 7638	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f ✚ 𝐵) = (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))))
56	55	3adant3 1132	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝐴 ∘f ✚ 𝐵) = (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))))
57	24	lmodfgrp 20804	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Grp)
58	57	grpmndd 18861	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Mnd)
59	58	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) → 𝑆 ∈ Mnd)
60		ffvelcdm 7020	. . . . . . . . . . . . . 14 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ 𝑅)
61	60	ex 412	. . . . . . . . . . . . 13 ⊢ (𝐴:𝑉⟶𝑅 → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
62	12, 61	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
63	62	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
64	63	imp 406	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ 𝑅)
65	24	fveq2i 6831	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀))
66	26, 65	eqtri 2756	. . . . . . . . . 10 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
67	64, 66	eleqtrdi 2843	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ (Base‘(Scalar‘𝑀)))
68		ffvelcdm 7020	. . . . . . . . . . . . . 14 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐵‘𝑦) ∈ 𝑅)
69	68, 66	eleqtrdi 2843	. . . . . . . . . . . . 13 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀)))
70	69	ex 412	. . . . . . . . . . . 12 ⊢ (𝐵:𝑉⟶𝑅 → (𝑦 ∈ 𝑉 → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))))
71	29, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → (𝑦 ∈ 𝑉 → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))))
72	71	ad2antll 729	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑦 ∈ 𝑉 → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))))
73	72	imp 406	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀)))
74	24	eqcomi 2742	. . . . . . . . . . 11 ⊢ (Scalar‘𝑀) = 𝑆
75	74	fveq2i 6831	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
76		lincsum.b	. . . . . . . . . 10 ⊢ ✚ = (+g‘𝑆)
77	75, 76	mndcl 18652	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ (𝐴‘𝑦) ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))) → ((𝐴‘𝑦) ✚ (𝐵‘𝑦)) ∈ (Base‘(Scalar‘𝑀)))
78	59, 67, 73, 77	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) → ((𝐴‘𝑦) ✚ (𝐵‘𝑦)) ∈ (Base‘(Scalar‘𝑀)))
79	78	fmpttd 7054	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))):𝑉⟶(Base‘(Scalar‘𝑀)))
80		fvex 6841	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
81		elmapg 8769	. . . . . . . 8 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))):𝑉⟶(Base‘(Scalar‘𝑀))))
82	80, 50, 81	sylancr 587	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))):𝑉⟶(Base‘(Scalar‘𝑀))))
83	79, 82	mpbird 257	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
84	83	3adant3 1132	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
85	56, 84	eqeltrd 2833	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝐴 ∘f ✚ 𝐵) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
86		lincval 48534	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴 ∘f ✚ 𝐵) ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐴 ∘f ✚ 𝐵)( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐵)‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
87	10, 85, 8, 86	syl3anc 1373	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → ((𝐴 ∘f ✚ 𝐵)( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐵)‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
88	51, 53	anim12i 613	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → (𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉))
89	88	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉))
90	89	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉))
91	50	anim1i 615	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑉))
92		fnfvof 7633	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑉)) → ((𝐴 ∘f ✚ 𝐵)‘𝑥) = ((𝐴‘𝑥) ✚ (𝐵‘𝑥)))
93	90, 91, 92	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → ((𝐴 ∘f ✚ 𝐵)‘𝑥) = ((𝐴‘𝑥) ✚ (𝐵‘𝑥)))
94	76	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → ✚ = (+g‘𝑆))
95	94	oveqd 7369	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → ((𝐴‘𝑥) ✚ (𝐵‘𝑥)) = ((𝐴‘𝑥)(+g‘𝑆)(𝐵‘𝑥)))
96	93, 95	eqtrd 2768	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → ((𝐴 ∘f ✚ 𝐵)‘𝑥) = ((𝐴‘𝑥)(+g‘𝑆)(𝐵‘𝑥)))
97	96	oveq1d 7367	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (((𝐴 ∘f ✚ 𝐵)‘𝑥)( ·𝑠 ‘𝑀)𝑥) = (((𝐴‘𝑥)(+g‘𝑆)(𝐵‘𝑥))( ·𝑠 ‘𝑀)𝑥))
98	9	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝑀 ∈ LMod)
99	98	adantr 480	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod)
100	15	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
101	100	imp 406	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
102	32	ad2antll 729	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
103	102	imp 406	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝐵‘𝑥) ∈ 𝑅)
104	21	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
105	104	imp 406	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑀))
106		eqid 2733	. . . . . . . . 9 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
107	24	fveq2i 6831	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘(Scalar‘𝑀))
108	1, 3, 106, 25, 66, 107	lmodvsdir 20821	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ ((𝐴‘𝑥) ∈ 𝑅 ∧ (𝐵‘𝑥) ∈ 𝑅 ∧ 𝑥 ∈ (Base‘𝑀))) → (((𝐴‘𝑥)(+g‘𝑆)(𝐵‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) + ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
109	99, 101, 103, 105, 108	syl13anc 1374	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (((𝐴‘𝑥)(+g‘𝑆)(𝐵‘𝑥))( ·𝑠 ‘𝑀)𝑥) = (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) + ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
110	97, 109	eqtrd 2768	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (((𝐴 ∘f ✚ 𝐵)‘𝑥)( ·𝑠 ‘𝑀)𝑥) = (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) + ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))
111	110	mpteq2dva 5186	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑥 ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐵)‘𝑥)( ·𝑠 ‘𝑀)𝑥)) = (𝑥 ∈ 𝑉 ↦ (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) + ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
112	111	oveq2d 7368	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑀 Σg (𝑥 ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐵)‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) + ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
113	112	3adant3 1132	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑀 Σg (𝑥 ∈ 𝑉 ↦ (((𝐴 ∘f ✚ 𝐵)‘𝑥)( ·𝑠 ‘𝑀)𝑥))) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) + ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
114	87, 113	eqtrd 2768	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → ((𝐴 ∘f ✚ 𝐵)( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ (((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) + ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
115		lincsum.x	. . . 4 ⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉)
116		lincsum.y	. . . 4 ⊢ 𝑌 = (𝐵( linC ‘𝑀)𝑉)
117	115, 116	oveq12i 7364	. . 3 ⊢ (𝑋 + 𝑌) = ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉))
118	66	oveq1i 7362	. . . . . . . . 9 ⊢ (𝑅 ↑m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
119	118	eleq2i 2825	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
120	119	biimpi 216	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
121	120	ad2antrl 728	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
122		lincval 48534	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
123	98, 121, 50, 122	syl3anc 1373	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
124	118	eleq2i 2825	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) ↔ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
125	124	biimpi 216	. . . . . . 7 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
126	125	ad2antll 729	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
127		lincval 48534	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐵( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
128	98, 126, 50, 127	syl3anc 1373	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐵( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥))))
129	123, 128	oveq12d 7370	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉)) = ((𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
130	129	3adant3 1132	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉)) = ((𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
131	117, 130	eqtrid 2780	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑋 + 𝑌) = ((𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥))) + (𝑀 Σg (𝑥 ∈ 𝑉 ↦ ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥)))))
132	49, 114, 131	3eqtr4rd 2779	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆))) → (𝑋 + 𝑌) = ((𝐴 ∘f ✚ 𝐵)( linC ‘𝑀)𝑉))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3437  𝒫 cpw 4549   class class class wbr 5093   ↦ cmpt 5174   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ∘_f cof 7614   ↑_m cmap 8756   finSupp cfsupp 9252  Basecbs 17122  +_gcplusg 17163  Scalarcsca 17166   ·_𝑠 cvsca 17167  0_gc0g 17345   Σ_g cgsu 17346  Mndcmnd 18644  CMndccmn 19694  LModclmod 20795   linC clinc 48529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-n0 12389  df-z 12476  df-uz 12739  df-fz 13410  df-fzo 13557  df-seq 13911  df-hash 14240  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-0g 17347  df-gsum 17348  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-submnd 18694  df-grp 18851  df-minusg 18852  df-cntz 19231  df-cmn 19696  df-abl 19697  df-mgp 20061  df-ur 20102  df-ring 20155  df-lmod 20797  df-linc 48531

This theorem is referenced by:  lincsumcl  48556