Theorem lincscmcl 49189

Description: The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)

Hypotheses

Ref	Expression
lincscmcl.s	⊢ · = ( ·𝑠 ‘𝑀)
lincscmcl.r	⊢ 𝑅 = (Base‘(Scalar‘𝑀))

Assertion

Ref	Expression
lincscmcl	⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))

Proof of Theorem lincscmcl

Dummy variables 𝑠 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2763	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscmcl.r	. . . . 5 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4	1, 2, 3	lcoval 49169	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))))
5	4	adantr 485	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))))
6		simpl 487	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
7	6	ad2antrr 738	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝑀 ∈ LMod)
8		simpr 489	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → 𝐶 ∈ 𝑅)
9	8	adantr 485	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐶 ∈ 𝑅)
10		simprl 782	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐷 ∈ (Base‘𝑀))
11		lincscmcl.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑀)
12	1, 2, 11, 3	lmodvscl 20980	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
13	7, 9, 10, 12	syl3anc 1398	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
14	2	lmodring 20970	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
15	14	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → (Scalar‘𝑀) ∈ Ring)
16	15	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (Scalar‘𝑀) ∈ Ring)
17	16	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) → (Scalar‘𝑀) ∈ Ring)
18	8	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → 𝐶 ∈ 𝑅)
19	18	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ 𝑅)
20		elmapi 8847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑅 ↑m 𝑉) → 𝑥:𝑉⟶𝑅)
21		ffvelcdm 7078	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥:𝑉⟶𝑅 ∧ 𝑣 ∈ 𝑉) → (𝑥‘𝑣) ∈ 𝑅)
22	21	ex 417	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥:𝑉⟶𝑅 → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
23	20, 22	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑅 ↑m 𝑉) → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
24	23	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
25	24	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
26	25	imp 411	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) → (𝑥‘𝑣) ∈ 𝑅)
27		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
28	3, 27	ringcl 20333	. . . . . . . . . . . . . 14 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ 𝐶 ∈ 𝑅 ∧ (𝑥‘𝑣) ∈ 𝑅) → (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)) ∈ 𝑅)
29	17, 19, 26, 28	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) → (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)) ∈ 𝑅)
30	29	fmpttd 7112	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))):𝑉⟶𝑅)
31	3	fvexi 6897	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ V
32		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
33	32	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → 𝑉 ∈ 𝒫 (Base‘𝑀))
34	33	adantl 486	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
35		elmapg 8837	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) ∈ (𝑅 ↑m 𝑉) ↔ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))):𝑉⟶𝑅))
36	31, 34, 35	sylancr 598	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) ∈ (𝑅 ↑m 𝑉) ↔ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))):𝑉⟶𝑅))
37	30, 36	mpbird 260	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) ∈ (𝑅 ↑m 𝑉))
38	15, 33, 8	3jca 1146	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶 ∈ 𝑅))
39	38	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶 ∈ 𝑅))
40		simpl 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ (𝑅 ↑m 𝑉))
41	40	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → 𝑥 ∈ (𝑅 ↑m 𝑉))
42		simprl 782	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
43	42	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
44	3	rmfsupp 49130	. . . . . . . . . . . 12 ⊢ ((((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶 ∈ 𝑅) ∧ 𝑥 ∈ (𝑅 ↑m 𝑉) ∧ 𝑥 finSupp (0g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0g‘(Scalar‘𝑀)))
45	39, 41, 43, 44	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0g‘(Scalar‘𝑀)))
46		oveq2 7420	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
47	46	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
48	47	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
49	48	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
50		simprl 782	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
51	40	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → 𝑥 ∈ (𝑅 ↑m 𝑉))
52	51, 8	anim12i 624	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝑥 ∈ (𝑅 ↑m 𝑉) ∧ 𝐶 ∈ 𝑅))
53		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
54		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) = (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))
55	11, 27, 53, 3, 54	lincscm 49187	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑥 ∈ (𝑅 ↑m 𝑉) ∧ 𝐶 ∈ 𝑅) ∧ 𝑥 finSupp (0g‘(Scalar‘𝑀))) → (𝐶 · (𝑥( linC ‘𝑀)𝑉)) = ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))
56	50, 52, 43, 55	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝐶 · (𝑥( linC ‘𝑀)𝑉)) = ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))
57	49, 56	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (𝐶 · 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))
58		breq1 5113	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0g‘(Scalar‘𝑀))))
59		oveq1 7419	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))
60	59	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) → ((𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉) ↔ (𝐶 · 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉)))
61	58, 60	anbi12d 643	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) → ((𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))))
62	61	rspcev 3582	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) ∈ (𝑅 ↑m 𝑉) ∧ ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))) → ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
63	37, 45, 57, 62	syl12anc 849	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
64	63	ex 417	. . . . . . . . 9 ⊢ (((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉))))
65	64	ex 417	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐷 ∈ (Base‘𝑀) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
66	65	rexlimiva 3158	. . . . . . 7 ⊢ (∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐷 ∈ (Base‘𝑀) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
67	66	impcom 412	. . . . . 6 ⊢ ((𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉))))
68	67	impcom 412	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
69	1, 2, 3	lcoval 49169	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 · 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
70	69	ad2antrr 738	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → ((𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 · 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ (𝑅 ↑m 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
71	13, 68, 70	mpbir2and 725	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))
72	71	ex 417	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ((𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑m 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)))
73	5, 72	sylbid 243	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)))
74	73	3impia 1135	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455  𝒫 cpw 4563   class class class wbr 5110   ↦ cmpt 5193  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ↑_m cmap 8825   finSupp cfsupp 9322  Basecbs 17270  ._rcmulr 17312  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17493  Ringcrg 20316  LModclmod 20962   linC clinc 49161   LinCo clinco 49162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-supp 8158  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-fsupp 9323  df-oi 9473  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-fzo 13685  df-seq 14040  df-hash 14369  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-plusg 17324  df-0g 17495  df-gsum 17496  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-mhm 18842  df-grp 19004  df-minusg 19005  df-ghm 19285  df-cntz 19388  df-cmn 19853  df-abl 19854  df-mgp 20218  df-rng 20232  df-ur 20265  df-ring 20318  df-lmod 20964  df-linc 49163  df-lco 49164

This theorem is referenced by:  lincsumscmcl  49190