Theorem lincsum 49485

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 2761	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2761	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
3		lincsum.p	. . 3 ⊢ + = (+g‘𝑀)
4		lmodcmn 21165	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
5	4	adantr 486	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ CMnd)
6	5	3ad2ant1 1151	. . 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 490	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
8	7	3ad2ant1 1151	. . 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 488	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
10	9	3ad2ant1 1151	. . . . 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 486	. . . 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 8853	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
13		ffvelcdm 7073	. . . . . . . . 9 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
14	13	ex 418	. . . . . . . 8 ⊢ (𝐴:𝑉⟶𝑅 → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
15	12, 14	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
16	15	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
17	16	3ad2ant2 1152	. . . . 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 412	. . . 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 4567	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
20	19	expcom 419	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
21	20	adantl 487	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
22	21	3ad2ant1 1151	. . . . 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 412	. . . 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 2761	. . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
26		lincsum.r	. . . . 5 ⊢ 𝑅 = (Base‘𝑆)
27	1, 24, 25, 26	lmodvscl 21133	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴‘𝑥) ∈ 𝑅 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
28	11, 18, 23, 27	syl3anc 1398	. . 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 8853	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵:𝑉⟶𝑅)
30		ffvelcdm 7073	. . . . . . . . 9 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐵‘𝑥) ∈ 𝑅)
31	30	ex 418	. . . . . . . 8 ⊢ (𝐵:𝑉⟶𝑅 → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
32	29, 31	syl 18	. . . . . . 7 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
33	32	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
34	33	3ad2ant2 1152	. . . . 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 412	. . . 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 21133	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐵‘𝑥) ∈ 𝑅 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
37	11, 35, 23, 36	syl3anc 1398	. . 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 2762	. . 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 2762	. . 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 23	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
41		simpl 488	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐴 ∈ (𝑅 ↑m 𝑉))
42		simpl 488	. . . 4 ⊢ ((𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆)) → 𝐴 finSupp (0g‘𝑆))
43	24, 26	scmfsupp 49431	. . . 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 1178	. . 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 490	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐵 ∈ (𝑅 ↑m 𝑉))
46		simpr 490	. . . 4 ⊢ ((𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆)) → 𝐵 finSupp (0g‘𝑆))
47	24, 26	scmfsupp 49431	. . . 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 1178	. . 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 20118	. 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 486	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
51		elmapfn 8871	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 Fn 𝑉)
52	51	ad2antrl 741	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐴 Fn 𝑉)
53		elmapfn 8871	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵 Fn 𝑉)
54	53	ad2antll 742	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵 Fn 𝑉)
55	50, 52, 54	offvalfv 7704	. . . . . 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 1150	. . . . 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 21124	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Grp)
58	57	grpmndd 19137	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Mnd)
59	58	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) → 𝑆 ∈ Mnd)
60		ffvelcdm 7073	. . . . . . . . . . . . . 14 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ 𝑅)
61	60	ex 418	. . . . . . . . . . . . 13 ⊢ (𝐴:𝑉⟶𝑅 → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
62	12, 61	syl 18	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
63	62	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
64	63	imp 412	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ 𝑅)
65	24	fveq2i 6880	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀))
66	26, 65	eqtri 2784	. . . . . . . . . 10 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
67	64, 66	eleqtrdi 2871	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ (Base‘(Scalar‘𝑀)))
68		ffvelcdm 7073	. . . . . . . . . . . . . 14 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐵‘𝑦) ∈ 𝑅)
69	68, 66	eleqtrdi 2871	. . . . . . . . . . . . 13 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀)))
70	69	ex 418	. . . . . . . . . . . 12 ⊢ (𝐵:𝑉⟶𝑅 → (𝑦 ∈ 𝑉 → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))))
71	29, 70	syl 18	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → (𝑦 ∈ 𝑉 → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))))
72	71	ad2antll 742	. . . . . . . . . 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 412	. . . . . . . . 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 2770	. . . . . . . . . . 11 ⊢ (Scalar‘𝑀) = 𝑆
75	74	fveq2i 6880	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
76		lincsum.b	. . . . . . . . . 10 ⊢ ✚ = (+g‘𝑆)
77	75, 76	mndcl 18911	. . . . . . . . 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 1398	. . . . . . . 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 7107	. . . . . . 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 6890	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
81		elmapg 8843	. . . . . . . 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 599	. . . . . . 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 260	. . . . . 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 1150	. . . . 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 2861	. . . 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 49465	. . . 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 1398	. . 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 625	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → (𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉))
89	88	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉))
90	89	adantr 486	. . . . . . . . . 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 627	. . . . . . . . . 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 7699	. . . . . . . . . 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 596	. . . . . . . . 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 7429	. . . . . . . . 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 2796	. . . . . . . 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 7427	. . . . . . 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 486	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝑀 ∈ LMod)
99	98	adantr 486	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod)
100	15	ad2antrl 741	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
101	100	imp 412	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
102	32	ad2antll 742	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
103	102	imp 412	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝐵‘𝑥) ∈ 𝑅)
104	21	adantr 486	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
105	104	imp 412	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑀))
106		eqid 2761	. . . . . . . . 9 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
107	24	fveq2i 6880	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘(Scalar‘𝑀))
108	1, 3, 106, 25, 66, 107	lmodvsdir 21141	. . . . . . . 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 1399	. . . . . . 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 2796	. . . . . 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 5198	. . . . 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 7428	. . . 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 1150	. . 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 2796	. 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 7424	. . 3 ⊢ (𝑋 + 𝑌) = ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉))
118	66	oveq1i 7422	. . . . . . . . 9 ⊢ (𝑅 ↑m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
119	118	eleq2i 2853	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
120	119	biimpi 219	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
121	120	ad2antrl 741	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
122		lincval 49465	. . . . . 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 1398	. . . . 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 2853	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) ↔ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
125	124	biimpi 219	. . . . . . 7 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
126	125	ad2antll 742	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
127		lincval 49465	. . . . . 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 1398	. . . . 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 7430	. . . 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 1150	. . 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 2808	. 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 2807	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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ∘_f cof 7680   ↑_m cmap 8831   finSupp cfsupp 9337  Basecbs 17367  +_gcplusg 17408  Scalarcsca 17411   ·_𝑠 cvsca 17412  0_gc0g 17590   Σ_g cgsu 17591  Mndcmnd 18903  CMndccmn 19974  LModclmod 21115   linC clinc 49460

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9338  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-n0 12588  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-seq 14125  df-hash 14455  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-0g 17592  df-gsum 17593  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-submnd 18959  df-grp 19127  df-minusg 19128  df-cntz 19511  df-cmn 19976  df-abl 19977  df-mgp 20341  df-ur 20388  df-ring 20441  df-lmod 21117  df-linc 49462

This theorem is used by:  lincsumcl  49487