Theorem lincscmcl 49759

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 49739	. . . 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 21186	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
13	7, 9, 10, 12	syl3anc 1399	. . . . 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 21176	. . . . . . . . . . . . . . . . 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 8900	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑅 ↑m 𝑉) → 𝑥:𝑉⟶𝑅)
21		ffvelcdm 7108	. . . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . 15 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
28	3, 27	ringcl 20510	. . . . . . . . . . . . . 14 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ 𝐶 ∈ 𝑅 ∧ (𝑥‘𝑣) ∈ 𝑅) → (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)) ∈ 𝑅)
29	17, 19, 26, 28	syl3anc 1399	. . . . . . . . . . . . 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 7142	. . . . . . . . . . . 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 6925	. . . . . . . . . . . . 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 8890	. . . . . . . . . . . . 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 49700	. . . . . . . . . . . 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 1399	. . . . . . . . . . 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 7454	. . . . . . . . . . . . . . 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 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 49757	. . . . . . . . . . . . 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 1399	. . . . . . . . . . . 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 5109	. . . . . . . . . . . . 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 7453	. . . . . . . . . . . . . 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 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 3580	. . . . . . . . . . 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 3159	. . . . . . 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 49739	. . . . . 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 2147  ∃wrex 3090  Vcvv 3454  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5190  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446   ↑_m cmap 8878   finSupp cfsupp 9384  Basecbs 17419  ._rcmulr 17461  Scalarcsca 17463   ·_𝑠 cvsca 17464  0_gc0g 17642  Ringcrg 20492  LModclmod 21168   linC clinc 49731   LinCo clinco 49732

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-supp 8208  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-fsupp 9385  df-oi 9534  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-n0 12637  df-z 12724  df-uz 12996  df-fz 13670  df-fzo 13819  df-seq 14175  df-hash 14505  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-plusg 17473  df-0g 17644  df-gsum 17645  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-mhm 19010  df-grp 19179  df-minusg 19180  df-ghm 19460  df-cntz 19564  df-cmn 20029  df-abl 20030  df-mgp 20394  df-rng 20408  df-ur 20441  df-ring 20494  df-lmod 21170  df-linc 49733  df-lco 49734

This theorem is used by:  lincsumscmcl  49760