Theorem lincscm 48276

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 2735	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2735	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscm.r	. . 3 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4		eqid 2735	. . 3 ⊢ (0g‘𝑀) = (0g‘𝑀)
5		eqid 2735	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
6		lincscm.s	. . 3 ⊢ ∙ = ( ·𝑠 ‘𝑀)
7		simp1l 1196	. . 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 1132	. . 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 1133	. . 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 8888	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅)
14		ffvelcdm 7101	. . . . . . . . 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 1133	. . . . 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 4615	. . . . . . . 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 1132	. . . . 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 2735	. . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
26	1, 2, 25, 3	lmodvscl 20893	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴‘𝑣) ∈ 𝑅 ∧ 𝑣 ∈ (Base‘𝑀)) → ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣) ∈ (Base‘𝑀))
27	12, 19, 24, 26	syl3anc 1370	. . 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 48220	. . . 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 1178	. . 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 20941	. 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 20883	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
32	31	adantr 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Ring)
33	32	3ad2ant1 1132	. . . . . . . 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 2831	. . . . . . . . . . 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 1133	. . . . . . . 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 7101	. . . . . . . . . . . . 13 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
41	40, 3	eleqtrdi 2849	. . . . . . . . . . . 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 1133	. . . . . . . 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 2735	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
48		lincscm.t	. . . . . . . 8 ⊢ · = (.r‘(Scalar‘𝑀))
49	47, 48	ringcl 20268	. . . . . . 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 1370	. . . . . 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 7134	. . . . 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 6920	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
54		elmapg 8878	. . . . . 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 48255	. . . 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 1370	. . 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 7464	. . . . . . . 8 ⊢ (𝑆 · (𝐴‘𝑣)) ∈ V
61		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝐴‘𝑥) = (𝐴‘𝑣))
62	61	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (𝑆 · (𝐴‘𝑥)) = (𝑆 · (𝐴‘𝑣)))
63	62, 51	fvmptg 7014	. . . . . . . 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 7446	. . . . . 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 20902	. . . . . . . 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 1371	. . . . . . 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 2744	. . . . . . . . 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 7448	. . . . . . 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 2775	. . . . . 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 2775	. . . . 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 5248	. . . 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 7447	. . 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 2775	. 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 7441	. . . . . . . . 9 ⊢ (𝑅 ↑m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑m 𝑉)
80	79	eleq2i 2831	. . . . . . . 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 1133	. . . . 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 48255	. . . . 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 1370	. . . 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 2775	. . 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 7447	. 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 2786	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 1537   ∈ wcel 2106  Vcvv 3478  𝒫 cpw 4605   class class class wbr 5148   ↦ cmpt 5231  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865   finSupp cfsupp 9399  Basecbs 17245  +_gcplusg 17298  ._rcmulr 17299  Scalarcsca 17301   ·_𝑠 cvsca 17302  0_gc0g 17486   Σ_g cgsu 17487  Ringcrg 20251  LModclmod 20875   linC clinc 48250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-0g 17488  df-gsum 17489  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-ur 20200  df-ring 20253  df-lmod 20877  df-linc 48252

This theorem is referenced by:  lincscmcl  48278