Theorem lincscm 49636

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 2760	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2760	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscm.r	. . 3 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4		eqid 2760	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
5		eqid 2760	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
6		lincscm.s	. . 3 ⊢ ∙ = ( ·𝑠 ‘𝑀)
7		simp1l 1216	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑀 ∈ LMod)
8		simpr 490	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
9	8	3ad2ant1 1151	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
10		simpr 490	. . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → 𝑆 ∈ 𝑅)
11	10	3ad2ant2 1152	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → 𝑆 ∈ 𝑅)
12	7	adantr 486	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
13		elmapi 8873	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
14		ffvelcdm 7082	. . . . . . . . 9 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑣 ∈ 𝑉) → (𝐴‘𝑣) ∈ 𝑅)
15	14	ex 418	. . . . . . . 8 ⊢ (𝐴:𝑉⟶𝑅 → (𝑣 ∈ 𝑉 → (𝐴‘𝑣) ∈ 𝑅))
16	13, 15	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑣 ∈ 𝑉 → (𝐴‘𝑣) ∈ 𝑅))
17	16	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → (𝑣 ∈ 𝑉 → (𝐴‘𝑣) ∈ 𝑅))
18	17	3ad2ant2 1152	. . . . 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 412	. . . 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 419	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
22	21	adantl 487	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
23	22	3ad2ant1 1151	. . . . 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 412	. . . 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 2760	. . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
26	1, 2, 25, 3	lmodvscl 21159	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴‘𝑣) ∈ 𝑅 ∧ 𝑣 ∈ (Base‘𝑀)) → ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ (Base‘𝑀))
27	12, 19, 24, 26	syl3anc 1398	. . 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 49581	. . . 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 1198	. . 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 21207	. 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 21149	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
32	31	adantr 486	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Ring)
33	32	3ad2ant1 1151	. . . . . . . 8 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (Scalar‘𝑀) ∈ Ring)
34	33	adantr 486	. . . . . . 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 2852	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑅 ↔ 𝑆 ∈ (Base‘(Scalar‘𝑀)))
36	35	biimpi 219	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑅 → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
37	36	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
38	37	3ad2ant2 1152	. . . . . . . 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 486	. . . . . . 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 7082	. . . . . . . . . . . . 13 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
41	40, 3	eleqtrdi 2870	. . . . . . . . . . . 12 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀)))
42	41	ex 418	. . . . . . . . . . 11 ⊢ (𝐴:𝑉⟶𝑅 → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))))
43	13, 42	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))))
44	43	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))))
45	44	3ad2ant2 1152	. . . . . . . 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 412	. . . . . . 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 2760	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
48		lincscm.t	. . . . . . . 8 ⊢ · = (.r‘(Scalar‘𝑀))
49	47, 48	ringcl 20483	. . . . . . 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 1398	. . . . . 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 7115	. . . . 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 6899	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
54		elmapg 8863	. . . . . 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 599	. . . . 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 260	. . . 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 49615	. . . 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 1398	. . 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 490	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
60		ovex 7454	. . . . . . . 8 ⊢ (𝑆 · (𝐴‘𝑣)) ∈ V
61		fveq2 6886	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝐴‘𝑥) = (𝐴‘𝑣))
62	61	oveq2d 7437	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑆 · (𝐴‘𝑥)) = (𝑆 · (𝐴‘𝑣)))
63	62, 51	fvmptg 6992	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ (𝑆 · (𝐴‘𝑣)) ∈ V) → (𝐹‘𝑣) = (𝑆 · (𝐴‘𝑣)))
64	59, 60, 63	sylancl 598	. . . . . . 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 7436	. . . . . 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 486	. . . . . . . 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 21168	. . . . . . . 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 1399	. . . . . . 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 2769	. . . . . . . . 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 7438	. . . . . . 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 2795	. . . . . 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 2795	. . . . 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 5197	. . . 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 7437	. . 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 2795	. 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 7431	. . . . . . . . 9 ⊢ (𝑅 ↑m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
80	79	eleq2i 2852	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
81	80	biimpi 219	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
82	81	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑m 𝑉))
83	82	3ad2ant2 1152	. . . . 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 49615	. . . . 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 1398	. . . 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 2795	. . 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 7437	. 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 2806	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 ∙ 𝑋) = (𝐹( linC ‘𝑀)𝑉))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450  𝒫 cpw 4556   class class class wbr 5102   ↦ cmpt 5185  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ↑_m cmap 8851   finSupp cfsupp 9357  Basecbs 17393  +_gcplusg 17434  ._rcmulr 17435  Scalarcsca 17437   ·_𝑠 cvsca 17438  0_gc0g 17616   Σ_g cgsu 17617  Ringcrg 20465  LModclmod 21141   linC clinc 49610

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-supp 8181  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-er 8721  df-map 8853  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-fsupp 9358  df-oi 9508  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-n0 12611  df-z 12698  df-uz 12970  df-fz 13644  df-fzo 13793  df-seq 14149  df-hash 14479  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-plusg 17447  df-0g 17618  df-gsum 17619  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-mhm 18984  df-grp 19153  df-minusg 19154  df-ghm 19434  df-cntz 19537  df-cmn 20002  df-abl 20003  df-mgp 20367  df-ur 20414  df-ring 20467  df-lmod 21143  df-linc 49612

This theorem is used by:  lincscmcl  49638