Theorem lincscm 48784

Description: A linear combinations multiplied with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.)

Hypotheses

Ref	Expression
lincscm.s	⊢ ∙ = ( ·𝑠 ‘𝑀)
lincscm.t	⊢ · = (.r‘(Scalar‘𝑀))
lincscm.x	⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉)
lincscm.r	⊢ 𝑅 = (Base‘(Scalar‘𝑀))
lincscm.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑆 · (𝐴‘𝑥)))

Assertion

Ref	Expression
lincscm	⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 ∙ 𝑋) = (𝐹( linC ‘𝑀)𝑉))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑀   𝑥,𝑅   𝑥,𝑆   𝑥,𝑉   𝑥, ·

Allowed substitution hints:   ∙ (𝑥)   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem lincscm

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2737	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscm.r	. . 3 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4		eqid 2737	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
5		eqid 2737	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
6		lincscm.s	. . 3 ⊢ ∙ = ( ·𝑠 ‘𝑀)
7		simp1l 1199	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑀 ∈ LMod)
8		simpr 484	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
9	8	3ad2ant1 1134	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
10		simpr 484	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → 𝑆 ∈ 𝑅)
11	10	3ad2ant2 1135	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑆 ∈ 𝑅)
12	7	adantr 480	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
13		elmapi 8798	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
14		ffvelcdm 7035	. . . . . . . . 9 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑣 ∈ 𝑉) → (𝐴‘𝑣) ∈ 𝑅)
15	14	ex 412	. . . . . . . 8 ⊢ (𝐴:𝑉⟶𝑅 → (𝑣 ∈ 𝑉 → (𝐴‘𝑣) ∈ 𝑅))
16	13, 15	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑣 ∈ 𝑉 → (𝐴‘𝑣) ∈ 𝑅))
17	16	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → (𝑣 ∈ 𝑉 → (𝐴‘𝑣) ∈ 𝑅))
18	17	3ad2ant2 1135	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 → (𝐴‘𝑣) ∈ 𝑅))
19	18	imp 406	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → (𝐴‘𝑣) ∈ 𝑅)
20		elelpwi 4566	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑣 ∈ (Base‘𝑀))
21	20	expcom 413	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
22	21	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
23	22	3ad2ant1 1134	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
24	23	imp 406	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (Base‘𝑀))
25		eqid 2737	. . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
26	1, 2, 25, 3	lmodvscl 20841	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴‘𝑣) ∈ 𝑅 ∧ 𝑣 ∈ (Base‘𝑀)) → ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ (Base‘𝑀))
27	12, 19, 24, 26	syl3anc 1374	. . 3 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ (Base‘𝑀))
28	2, 3	scmfsupp 48729	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀))
29	28	3adant2r 1181	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀))
30	1, 2, 3, 4, 5, 6, 7, 9, 11, 27, 29	gsumvsmul 20889	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ (𝑆 ∙ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))) = (𝑆 ∙ (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))))
31	2	lmodring 20831	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
32	31	adantr 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Ring)
33	32	3ad2ant1 1134	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (Scalar‘𝑀) ∈ Ring)
34	33	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑥 ∈ 𝑉) → (Scalar‘𝑀) ∈ Ring)
35	3	eleq2i 2829	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑅 ↔ 𝑆 ∈ (Base‘(Scalar‘𝑀)))
36	35	biimpi 216	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑅 → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
37	36	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
38	37	3ad2ant2 1135	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
39	38	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑥 ∈ 𝑉) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
40		ffvelcdm 7035	. . . . . . . . . . . . 13 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
41	40, 3	eleqtrdi 2847	. . . . . . . . . . . 12 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀)))
42	41	ex 412	. . . . . . . . . . 11 ⊢ (𝐴:𝑉⟶𝑅 → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))))
43	13, 42	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))))
44	43	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))))
45	44	3ad2ant2 1135	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))))
46	45	imp 406	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀)))
47		eqid 2737	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
48		lincscm.t	. . . . . . . 8 ⊢ · = (.r‘(Scalar‘𝑀))
49	47, 48	ringcl 20197	. . . . . . 7 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ 𝑆 ∈ (Base‘(Scalar‘𝑀)) ∧ (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))) → (𝑆 · (𝐴‘𝑥)) ∈ (Base‘(Scalar‘𝑀)))
50	34, 39, 46, 49	syl3anc 1374	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑥 ∈ 𝑉) → (𝑆 · (𝐴‘𝑥)) ∈ (Base‘(Scalar‘𝑀)))
51		lincscm.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑆 · (𝐴‘𝑥)))
52	50, 51	fmptd 7068	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
53		fvex 6855	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
54		elmapg 8788	. . . . . 6 ⊢ (((Base‘(Scalar‘𝑀)) ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
55	53, 9, 54	sylancr 588	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ↔ 𝐹:𝑉⟶(Base‘(Scalar‘𝑀))))
56	52, 55	mpbird 257	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
57		lincval 48763	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
58	7, 56, 9, 57	syl3anc 1374	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
59		simpr 484	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
60		ovex 7401	. . . . . . . 8 ⊢ (𝑆 · (𝐴‘𝑣)) ∈ V
61		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝐴‘𝑥) = (𝐴‘𝑣))
62	61	oveq2d 7384	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑆 · (𝐴‘𝑥)) = (𝑆 · (𝐴‘𝑣)))
63	62, 51	fvmptg 6947	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ (𝑆 · (𝐴‘𝑣)) ∈ V) → (𝐹‘𝑣) = (𝑆 · (𝐴‘𝑣)))
64	59, 60, 63	sylancl 587	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = (𝑆 · (𝐴‘𝑣)))
65	64	oveq1d 7383	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = ((𝑆 · (𝐴‘𝑣))( ·𝑠 ‘𝑀)𝑣))
66	11	adantr 480	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑆 ∈ 𝑅)
67	1, 2, 25, 3, 48	lmodvsass 20850	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ (𝑆 ∈ 𝑅 ∧ (𝐴‘𝑣) ∈ 𝑅 ∧ 𝑣 ∈ (Base‘𝑀))) → ((𝑆 · (𝐴‘𝑣))( ·𝑠 ‘𝑀)𝑣) = (𝑆( ·𝑠 ‘𝑀)((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
68	12, 66, 19, 24, 67	syl13anc 1375	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝑆 · (𝐴‘𝑣))( ·𝑠 ‘𝑀)𝑣) = (𝑆( ·𝑠 ‘𝑀)((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
69	6	eqcomi 2746	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑀) = ∙
70	69	a1i 11	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ( ·𝑠 ‘𝑀) = ∙ )
71	70	oveqd 7385	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → (𝑆( ·𝑠 ‘𝑀)((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑆 ∙ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
72	68, 71	eqtrd 2772	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝑆 · (𝐴‘𝑣))( ·𝑠 ‘𝑀)𝑣) = (𝑆 ∙ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
73	65, 72	eqtrd 2772	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣) = (𝑆 ∙ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))
74	73	mpteq2dva 5193	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ (𝑆 ∙ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
75	74	oveq2d 7384	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ (𝑆 ∙ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))))
76	58, 75	eqtrd 2772	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ (𝑆 ∙ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))))
77		lincscm.x	. . . . 5 ⊢ 𝑋 = (𝐴( linC ‘𝑀)𝑉)
78	77	a1i 11	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑋 = (𝐴( linC ‘𝑀)𝑉))
79	3	oveq1i 7378	. . . . . . . . 9 ⊢ (𝑅 ↑m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
80	79	eleq2i 2829	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
81	80	biimpi 216	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
82	81	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
83	82	3ad2ant2 1135	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
84		lincval 48763	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
85	7, 83, 9, 84	syl3anc 1374	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
86	78, 85	eqtrd 2772	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣))))
87	86	oveq2d 7384	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 ∙ 𝑋) = (𝑆 ∙ (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)))))
88	30, 76, 87	3eqtr4rd 2783	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 ∙ 𝑋) = (𝐹( linC ‘𝑀)𝑉))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442  𝒫 cpw 4556   class class class wbr 5100   ↦ cmpt 5181  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775   finSupp cfsupp 9276  Basecbs 17148  +_gcplusg 17189  ._rcmulr 17190  Scalarcsca 17192   ·_𝑠 cvsca 17193  0_gc0g 17371   Σ_g cgsu 17372  Ringcrg 20180  LModclmod 20823   linC clinc 48758

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583  df-seq 13937  df-hash 14266  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-plusg 17202  df-0g 17373  df-gsum 17374  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-grp 18878  df-minusg 18879  df-ghm 19154  df-cntz 19258  df-cmn 19723  df-abl 19724  df-mgp 20088  df-ur 20129  df-ring 20182  df-lmod 20825  df-linc 48760

This theorem is referenced by:  lincscmcl  48786