Theorem lincscmcl 49488

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 2761	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2761	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscmcl.r	. . . . 5 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4	1, 2, 3	lcoval 49468	. . . 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 486	. . 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 488	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
7	6	ad2antrr 739	. . . . . 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 490	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → 𝐶 ∈ 𝑅)
9	8	adantr 486	. . . . . 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 783	. . . . . 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 21133	. . . . . 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 21123	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
15	14	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → (Scalar‘𝑀) ∈ Ring)
16	15	adantl 487	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 8853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑅 ↑m 𝑉) → 𝑥:𝑉⟶𝑅)
21		ffvelcdm 7073	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥:𝑉⟶𝑅 ∧ 𝑣 ∈ 𝑉) → (𝑥‘𝑣) ∈ 𝑅)
22	21	ex 418	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
25	24	ad2antrr 739	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . 15 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
28	3, 27	ringcl 20457	. . . . . . . . . . . . . 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 7107	. . . . . . . . . . . 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 6891	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ V
32		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
33	32	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → 𝑉 ∈ 𝒫 (Base‘𝑀))
34	33	adantl 487	. . . . . . . . . . . . 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 8843	. . . . . . . . . . . . 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 599	. . . . . . . . . . . 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 487	. . . . . . . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ (𝑅 ↑m 𝑉))
41	40	ad2antrr 739	. . . . . . . . . . . 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 783	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
43	42	ad2antrr 739	. . . . . . . . . . . 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 49429	. . . . . . . . . . . 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 487	. . . . . . . . . . . . . 14 ⊢ ((𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
48	47	adantl 487	. . . . . . . . . . . . 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 739	. . . . . . . . . . . 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 783	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 625	. . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . 14 ⊢ (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
54		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) = (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))
55	11, 27, 53, 3, 54	lincscm 49486	. . . . . . . . . . . . 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 2796	. . . . . . . . . . 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 5106	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 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 644	. . . . . . . . . . . 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 3577	. . . . . . . . . . 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 850	. . . . . . . . . 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 418	. . . . . . . . 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 418	. . . . . . . 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 3156	. . . . . . 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 413	. . . . . 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 413	. . . . 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 49468	. . . . . 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 739	. . . . 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 726	. . . 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 418	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  Vcvv 3451  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831   finSupp cfsupp 9337  Basecbs 17367  ._rcmulr 17409  Scalarcsca 17411   ·_𝑠 cvsca 17412  0_gc0g 17590  Ringcrg 20439  LModclmod 21115   linC clinc 49460   LinCo clinco 49461

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9338  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-n0 12588  df-z 12675  df-uz 12947  df-fz 13621  df-fzo 13769  df-seq 14125  df-hash 14455  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-plusg 17421  df-0g 17592  df-gsum 17593  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-mhm 18958  df-grp 19127  df-minusg 19128  df-ghm 19408  df-cntz 19511  df-cmn 19976  df-abl 19977  df-mgp 20341  df-rng 20355  df-ur 20388  df-ring 20441  df-lmod 21117  df-linc 49462  df-lco 49463

This theorem is used by:  lincsumscmcl  49489