Theorem lincsum 42012

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 2610	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2610	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
3		lincsum.p	. . 3 ⊢ + = (+g‘𝑀)
4		lmodcmn 18734	. . . . 5 ⊢ (𝑀 ∈ LMod → 𝑀 ∈ CMnd)
5	4	adantr 480	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ CMnd)
6	5	3ad2ant1 1075	. . 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 476	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
8	7	3ad2ant1 1075	. . 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 472	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
10	9	3ad2ant1 1075	. . . . 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 7765	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) → 𝐴:𝑉⟶𝑅)
13		ffvelrn 6265	. . . . . . . . 9 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
14	13	ex 449	. . . . . . . 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 1076	. . . . 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 444	. . . 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 4119	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑀))
20	19	expcom 450	. . . . . . 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 1075	. . . . 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 444	. . . 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 2610	. . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
26		lincsum.r	. . . . 5 ⊢ 𝑅 = (Base‘𝑆)
27	1, 24, 25, 26	lmodvscl 18703	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴‘𝑥) ∈ 𝑅 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝐴‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
28	11, 18, 23, 27	syl3anc 1318	. . 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 7765	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑𝑚 𝑉) → 𝐵:𝑉⟶𝑅)
30		ffvelrn 6265	. . . . . . . . 9 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐵‘𝑥) ∈ 𝑅)
31	30	ex 449	. . . . . . . 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 1076	. . . . 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 444	. . . 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 18703	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐵‘𝑥) ∈ 𝑅 ∧ 𝑥 ∈ (Base‘𝑀)) → ((𝐵‘𝑥)( ·𝑠 ‘𝑀)𝑥) ∈ (Base‘𝑀))
37	11, 35, 23, 36	syl3anc 1318	. . 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 2611	. . 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 2611	. . 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 472	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉)) → 𝐴 ∈ (𝑅 ↑𝑚 𝑉))
42		simpl 472	. . . 4 ⊢ ((𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆)) → 𝐴 finSupp (0g‘𝑆))
43	24, 26	scmfsupp 41953	. . . 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 1360	. . 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 476	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉)) → 𝐵 ∈ (𝑅 ↑𝑚 𝑉))
46		simpr 476	. . . 4 ⊢ ((𝐴 finSupp (0g‘𝑆) ∧ 𝐵 finSupp (0g‘𝑆)) → 𝐵 finSupp (0g‘𝑆))
47	24, 26	scmfsupp 41953	. . . 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 1360	. . 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 18147	. 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 7766	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) → 𝐴 Fn 𝑉)
52	51	ad2antrl 760	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → 𝐴 Fn 𝑉)
53		elmapfn 7766	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑𝑚 𝑉) → 𝐵 Fn 𝑉)
54	53	ad2antll 761	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → 𝐵 Fn 𝑉)
55	50, 52, 54	offvalfv 41914	. . . . . 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 1074	. . . . 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 18695	. . . . . . . . . . 11 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Grp)
58		grpmnd 17252	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
59	57, 58	syl 17	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → 𝑆 ∈ Mnd)
60	59	ad3antrrr 762	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑦 ∈ 𝑉) → 𝑆 ∈ Mnd)
61		ffvelrn 6265	. . . . . . . . . . . . . 14 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ 𝑅)
62	61	ex 449	. . . . . . . . . . . . 13 ⊢ (𝐴:𝑉⟶𝑅 → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
63	12, 62	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
64	63	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → (𝑦 ∈ 𝑉 → (𝐴‘𝑦) ∈ 𝑅))
65	64	imp 444	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ 𝑅)
66	24	fveq2i 6106	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘(Scalar‘𝑀))
67	26, 66	eqtri 2632	. . . . . . . . . 10 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
68	65, 67	syl6eleq 2698	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑦 ∈ 𝑉) → (𝐴‘𝑦) ∈ (Base‘(Scalar‘𝑀)))
69		ffvelrn 6265	. . . . . . . . . . . . . 14 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐵‘𝑦) ∈ 𝑅)
70	69, 67	syl6eleq 2698	. . . . . . . . . . . . 13 ⊢ ((𝐵:𝑉⟶𝑅 ∧ 𝑦 ∈ 𝑉) → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀)))
71	70	ex 449	. . . . . . . . . . . 12 ⊢ (𝐵:𝑉⟶𝑅 → (𝑦 ∈ 𝑉 → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))))
72	29, 71	syl 17	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (𝑅 ↑𝑚 𝑉) → (𝑦 ∈ 𝑉 → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))))
73	72	ad2antll 761	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → (𝑦 ∈ 𝑉 → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))))
74	73	imp 444	. . . . . . . . 9 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑦 ∈ 𝑉) → (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀)))
75	24	eqcomi 2619	. . . . . . . . . . 11 ⊢ (Scalar‘𝑀) = 𝑆
76	75	fveq2i 6106	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘𝑆)
77		lincsum.b	. . . . . . . . . 10 ⊢ ✚ = (+g‘𝑆)
78	76, 77	mndcl 17124	. . . . . . . . 9 ⊢ ((𝑆 ∈ Mnd ∧ (𝐴‘𝑦) ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐵‘𝑦) ∈ (Base‘(Scalar‘𝑀))) → ((𝐴‘𝑦) ✚ (𝐵‘𝑦)) ∈ (Base‘(Scalar‘𝑀)))
79	60, 68, 74, 78	syl3anc 1318	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑦 ∈ 𝑉) → ((𝐴‘𝑦) ✚ (𝐵‘𝑦)) ∈ (Base‘(Scalar‘𝑀)))
80		eqid 2610	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦))) = (𝑦 ∈ 𝑉 ↦ ((𝐴‘𝑦) ✚ (𝐵‘𝑦)))
81	79, 80	fmptd 6292	. . . . . . 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)))
82		fvex 6113	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
83		elmapg 7757	. . . . . . . 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))))
84	82, 50, 83	sylancr 694	. . . . . . 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))))
85	81, 84	mpbird 246	. . . . . 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))
86	85	3adant3 1074	. . . . 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))
87	56, 86	eqeltrd 2688	. . . 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))
88		lincval 41992	. . . 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))))
89	10, 87, 8, 88	syl3anc 1318	. . 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))))
90	51, 53	anim12i 588	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉)) → (𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉))
91	90	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → (𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉))
92	91	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))
93	50	anim1i 590	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑉))
94		fnfvof 6809	. . . . . . . . . 10 ⊢ (((𝐴 Fn 𝑉 ∧ 𝐵 Fn 𝑉) ∧ (𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝑥 ∈ 𝑉)) → ((𝐴 ∘𝑓 ✚ 𝐵)‘𝑥) = ((𝐴‘𝑥) ✚ (𝐵‘𝑥)))
95	92, 93, 94	syl2anc 691	. . . . . . . . 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)))
96	77	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑥 ∈ 𝑉) → ✚ = (+g‘𝑆))
97	96	oveqd 6566	. . . . . . . . 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)))
98	95, 97	eqtrd 2644	. . . . . . . 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)))
99	98	oveq1d 6564	. . . . . . 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))
100	9	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → 𝑀 ∈ LMod)
101	100	adantr 480	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑥 ∈ 𝑉) → 𝑀 ∈ LMod)
102	15	ad2antrl 760	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ 𝑅))
103	102	imp 444	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
104	32	ad2antll 761	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → (𝑥 ∈ 𝑉 → (𝐵‘𝑥) ∈ 𝑅))
105	104	imp 444	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑥 ∈ 𝑉) → (𝐵‘𝑥) ∈ 𝑅)
106	21	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → (𝑥 ∈ 𝑉 → 𝑥 ∈ (Base‘𝑀)))
107	106	imp 444	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑀))
108		eqid 2610	. . . . . . . . 9 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
109	24	fveq2i 6106	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘(Scalar‘𝑀))
110	1, 3, 108, 25, 67, 109	lmodvsdir 18710	. . . . . . . 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)))
111	101, 103, 105, 107, 110	syl13anc 1320	. . . . . . 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)))
112	99, 111	eqtrd 2644	. . . . . 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)))
113	112	mpteq2dva 4672	. . . . 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))))
114	113	oveq2d 6565	. . . 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)))))
115	114	3adant3 1074	. . 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)))))
116	89, 115	eqtrd 2644	. 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)))))
117		lincsum.x	. . . 4 ⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉)
118		lincsum.y	. . . 4 ⊢ 𝑌 = (𝐵( linC ‘𝑀)𝑉)
119	117, 118	oveq12i 6561	. . 3 ⊢ (𝑋 + 𝑌) = ((𝐴( linC ‘𝑀)𝑉) + (𝐵( linC ‘𝑀)𝑉))
120	67	oveq1i 6559	. . . . . . . . 9 ⊢ (𝑅 ↑𝑚 𝑉) = ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)
121	120	eleq2i 2680	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
122	121	biimpi 205	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
123	122	ad2antrl 760	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
124		lincval 41992	. . . . . 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))))
125	100, 123, 50, 124	syl3anc 1318	. . . . 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))))
126	120	eleq2i 2680	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑅 ↑𝑚 𝑉) ↔ 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
127	126	biimpi 205	. . . . . . 7 ⊢ (𝐵 ∈ (𝑅 ↑𝑚 𝑉) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
128	127	ad2antll 761	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → 𝐵 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
129		lincval 41992	. . . . . 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))))
130	100, 128, 50, 129	syl3anc 1318	. . . . 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))))
131	125, 130	oveq12d 6567	. . . 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)))))
132	131	3adant3 1074	. . 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)))))
133	119, 132	syl5eq 2656	. 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)))))
134	49, 116, 133	3eqtr4rd 2655	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583   ↦ cmpt 4643   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ↑_𝑚 cmap 7744   finSupp cfsupp 8158  Basecbs 15695  +_gcplusg 15768  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923   Σ_g cgsu 15924  Mndcmnd 17117  Grpcgrp 17245  CMndccmn 18016  LModclmod 18686   linC clinc 41987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-linc 41989

This theorem is referenced by:  lincsumcl  42014