lincscm - Mathbox for Alexander van der Vekens

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Theorem lincscm 47013

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 (0_g‘(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 2733	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2733	. . 3 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscm.r	. . 3 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4		eqid 2733	. . 3 ⊢ (0_g‘𝑀) = (0_g‘𝑀)
5		eqid 2733	. . 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 (0_g‘(Scalar‘𝑀))) → 𝑀 ∈ LMod)
8		simpr 486	. . . 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 (0_g‘(Scalar‘𝑀))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
10		simpr 486	. . . 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 (0_g‘(Scalar‘𝑀))) → 𝑆 ∈ 𝑅)
12	7	adantr 482	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑀 ∈ LMod)
13		elmapi 8839	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑_m 𝑉) → 𝐴:𝑉⟶𝑅)
14		ffvelcdm 7079	. . . . . . . . 9 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑣 ∈ 𝑉) → (𝐴‘𝑣) ∈ 𝑅)
15	14	ex 414	. . . . . . . 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 482	. . . . . 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 (0_g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 → (𝐴‘𝑣) ∈ 𝑅))
19	18	imp 408	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → (𝐴‘𝑣) ∈ 𝑅)
20		elelpwi 4611	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑣 ∈ (Base‘𝑀))
21	20	expcom 415	. . . . . . 7 ⊢ (𝑉 ∈ 𝒫 (Base‘𝑀) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
22	21	adantl 483	. . . . . 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 (0_g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 → 𝑣 ∈ (Base‘𝑀)))
24	23	imp 408	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ (Base‘𝑀))
25		eqid 2733	. . . . 5 ⊢ ( ·_𝑠 ‘𝑀) = ( ·_𝑠 ‘𝑀)
26	1, 2, 25, 3	lmodvscl 20477	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ (𝐴‘𝑣) ∈ 𝑅 ∧ 𝑣 ∈ (Base‘𝑀)) → ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ∈ (Base‘𝑀))
27	12, 19, 24, 26	syl3anc 1372	. . 3 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣) ∈ (Base‘𝑀))
28	2, 3	scmfsupp 46956	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) finSupp (0_g‘𝑀))
29	28	3adant2r 1180	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) finSupp (0_g‘𝑀))
30	1, 2, 3, 4, 5, 6, 7, 9, 11, 27, 29	gsumvsmul 20524	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ (𝑆 ∙ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)))) = (𝑆 ∙ (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)))))
31	2	lmodring 20467	. . . . . . . . . 10 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
32	31	adantr 482	. . . . . . . . 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 (0_g‘(Scalar‘𝑀))) → (Scalar‘𝑀) ∈ Ring)
34	33	adantr 482	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑥 ∈ 𝑉) → (Scalar‘𝑀) ∈ Ring)
35	3	eleq2i 2826	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑅 ↔ 𝑆 ∈ (Base‘(Scalar‘𝑀)))
36	35	biimpi 215	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑅 → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
37	36	adantl 483	. . . . . . . . 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 (0_g‘(Scalar‘𝑀))) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
39	38	adantr 482	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑥 ∈ 𝑉) → 𝑆 ∈ (Base‘(Scalar‘𝑀)))
40		ffvelcdm 7079	. . . . . . . . . . . . 13 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ 𝑅)
41	40, 3	eleqtrdi 2844	. . . . . . . . . . . 12 ⊢ ((𝐴:𝑉⟶𝑅 ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀)))
42	41	ex 414	. . . . . . . . . . 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 482	. . . . . . . . 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 (0_g‘(Scalar‘𝑀))) → (𝑥 ∈ 𝑉 → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀))))
46	45	imp 408	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑥 ∈ 𝑉) → (𝐴‘𝑥) ∈ (Base‘(Scalar‘𝑀)))
47		eqid 2733	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
48		lincscm.t	. . . . . . . 8 ⊢ · = (._r‘(Scalar‘𝑀))
49	47, 48	ringcl 20064	. . . . . . 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 1372	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑥 ∈ 𝑉) → (𝑆 · (𝐴‘𝑥)) ∈ (Base‘(Scalar‘𝑀)))
51		lincscm.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ (𝑆 · (𝐴‘𝑥)))
52	50, 51	fmptd 7109	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → 𝐹:𝑉⟶(Base‘(Scalar‘𝑀)))
53		fvex 6901	. . . . . 6 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
54		elmapg 8829	. . . . . 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 (0_g‘(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 (0_g‘(Scalar‘𝑀))) → 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
57		lincval 46992	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣))))
58	7, 56, 9, 57	syl3anc 1372	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣))))
59		simpr 486	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
60		ovex 7437	. . . . . . . 8 ⊢ (𝑆 · (𝐴‘𝑣)) ∈ V
61		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝐴‘𝑥) = (𝐴‘𝑣))
62	61	oveq2d 7420	. . . . . . . . 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 587	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → (𝐹‘𝑣) = (𝑆 · (𝐴‘𝑣)))
65	64	oveq1d 7419	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) = ((𝑆 · (𝐴‘𝑣))( ·_𝑠 ‘𝑀)𝑣))
66	11	adantr 482	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → 𝑆 ∈ 𝑅)
67	1, 2, 25, 3, 48	lmodvsass 20485	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝑆 · (𝐴‘𝑣))( ·_𝑠 ‘𝑀)𝑣) = (𝑆( ·_𝑠 ‘𝑀)((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)))
69	6	eqcomi 2742	. . . . . . . . 9 ⊢ ( ·_𝑠 ‘𝑀) = ∙
70	69	a1i 11	. . . . . . . 8 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ( ·_𝑠 ‘𝑀) = ∙ )
71	70	oveqd 7421	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → (𝑆( ·_𝑠 ‘𝑀)((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) = (𝑆 ∙ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)))
72	68, 71	eqtrd 2773	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝑆 · (𝐴‘𝑣))( ·_𝑠 ‘𝑀)𝑣) = (𝑆 ∙ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)))
73	65, 72	eqtrd 2773	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) ∧ 𝑣 ∈ 𝑉) → ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣) = (𝑆 ∙ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)))
74	73	mpteq2dva 5247	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣)) = (𝑣 ∈ 𝑉 ↦ (𝑆 ∙ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣))))
75	74	oveq2d 7420	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·_𝑠 ‘𝑀)𝑣))) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ (𝑆 ∙ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)))))
76	58, 75	eqtrd 2773	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) = (𝑀 Σ_g (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 (0_g‘(Scalar‘𝑀))) → 𝑋 = (𝐴( linC ‘𝑀)𝑉))
79	3	oveq1i 7414	. . . . . . . . 9 ⊢ (𝑅 ↑_m 𝑉) = ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)
80	79	eleq2i 2826	. . . . . . . 8 ⊢ (𝐴 ∈ (𝑅 ↑_m 𝑉) ↔ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
81	80	biimpi 215	. . . . . . 7 ⊢ (𝐴 ∈ (𝑅 ↑_m 𝑉) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
82	81	adantr 482	. . . . . 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 (0_g‘(Scalar‘𝑀))) → 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
84		lincval 46992	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝐴 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣))))
85	7, 83, 9, 84	syl3anc 1372	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝐴( linC ‘𝑀)𝑉) = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣))))
86	78, 85	eqtrd 2773	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → 𝑋 = (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣))))
87	86	oveq2d 7420	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝑆 ∙ 𝑋) = (𝑆 ∙ (𝑀 Σ_g (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·_𝑠 ‘𝑀)𝑣)))))
88	30, 76, 87	3eqtr4rd 2784	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑆 ∈ 𝑅) ∧ 𝐴 finSupp (0_g‘(Scalar‘𝑀))) → (𝑆 ∙ 𝑋) = (𝐹( linC ‘𝑀)𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 𝒫 cpw 4601 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ↑_m cmap 8816 finSupp cfsupp 9357 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 Σ_g cgsu 17382 Ringcrg 20047 LModclmod 20459 linC clinc 46987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-0g 17383 df-gsum 17384 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-ghm 19084 df-cntz 19175 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-lmod 20461 df-linc 46989

This theorem is referenced by: lincscmcl 47015

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