Theorem lincscm 48461

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 2731	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2731	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscm.r	. . 3 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4		eqid 2731	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
5		eqid 2731	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
6		lincscm.s	. . 3 ⊢ ∙ = ( ·𝑠 ‘𝑀)
7		simp1l 1198	. . 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 1133	. . 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 1134	. . 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 8773	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
14		ffvelcdm 7014	. . . . . . . . 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 1134	. . . . 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 4560	. . . . . . . 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 1133	. . . . 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 2731	. . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
26	1, 2, 25, 3	lmodvscl 20809	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴‘𝑣) ∈ 𝑅 ∧ 𝑣 ∈ (Base‘𝑀)) → ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ (Base‘𝑀))
27	12, 19, 24, 26	syl3anc 1373	. . 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 48405	. . . 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 1180	. . 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 20857	. 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 20799	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
32	31	adantr 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Ring)
33	32	3ad2ant1 1133	. . . . . . . 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 2823	. . . . . . . . . . 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 1134	. . . . . . . 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 7014	. . . . . . . . . . . . 13 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
41	40, 3	eleqtrdi 2841	. . . . . . . . . . . 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 1134	. . . . . . . 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 2731	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
48		lincscm.t	. . . . . . . 8 ⊢ · = (.r‘(Scalar‘𝑀))
49	47, 48	ringcl 20166	. . . . . . 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 1373	. . . . . 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 7047	. . . . 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 6835	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
54		elmapg 8763	. . . . . 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 587	. . . . 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 48440	. . . 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 1373	. . 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 7379	. . . . . . . 8 ⊢ (𝑆 · (𝐴‘𝑣)) ∈ V
61		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝐴‘𝑥) = (𝐴‘𝑣))
62	61	oveq2d 7362	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑆 · (𝐴‘𝑥)) = (𝑆 · (𝐴‘𝑣)))
63	62, 51	fvmptg 6927	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ (𝑆 · (𝐴‘𝑣)) ∈ V) → (𝐹‘𝑣) = (𝑆 · (𝐴‘𝑣)))
64	59, 60, 63	sylancl 586	. . . . . . 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 7361	. . . . . 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 20818	. . . . . . . 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 1374	. . . . . . 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 2740	. . . . . . . . 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 7363	. . . . . . 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 2766	. . . . . 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 2766	. . . . 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 5184	. . . 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 7362	. . 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 2766	. 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 7356	. . . . . . . . 9 ⊢ (𝑅 ↑m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
80	79	eleq2i 2823	. . . . . . . 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 1134	. . . . 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 48440	. . . . 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 1373	. . . 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 2766	. . 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 7362	. 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 2777	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 1086   = wceq 1541   ∈ wcel 2111  Vcvv 3436  𝒫 cpw 4550   class class class wbr 5091   ↦ cmpt 5172  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750   finSupp cfsupp 9245  Basecbs 17117  +_gcplusg 17158  ._rcmulr 17159  Scalarcsca 17161   ·_𝑠 cvsca 17162  0_gc0g 17340   Σ_g cgsu 17341  Ringcrg 20149  LModclmod 20791   linC clinc 48435

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-fsupp 9246  df-oi 9396  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-n0 12379  df-z 12466  df-uz 12730  df-fz 13405  df-fzo 13552  df-seq 13906  df-hash 14235  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-plusg 17171  df-0g 17342  df-gsum 17343  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-mhm 18688  df-grp 18846  df-minusg 18847  df-ghm 19123  df-cntz 19227  df-cmn 19692  df-abl 19693  df-mgp 20057  df-ur 20098  df-ring 20151  df-lmod 20793  df-linc 48437

This theorem is referenced by:  lincscmcl  48463