Theorem lincscmcl 46025

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 2737	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2737	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscmcl.r	. . . . 5 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4	1, 2, 3	lcoval 46005	. . . 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 481	. . 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 483	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
7	6	ad2antrr 723	. . . . . 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 485	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → 𝐶 ∈ 𝑅)
9	8	adantr 481	. . . . . 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 768	. . . . . 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 20212	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
13	7, 9, 10, 12	syl3anc 1370	. . . . 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 20203	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
15	14	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → (Scalar‘𝑀) ∈ Ring)
16	15	adantl 482	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 8685	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑅 ↑m 𝑉) → 𝑥:𝑉⟶𝑅)
21		ffvelcdm 6998	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥:𝑉⟶𝑅 ∧ 𝑣 ∈ 𝑉) → (𝑥‘𝑣) ∈ 𝑅)
22	21	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥:𝑉⟶𝑅 → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
23	20, 22	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑅 ↑m 𝑉) → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
24	23	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
25	24	ad2antrr 723	. . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (.r‘(Scalar‘𝑀)) = (.r‘(Scalar‘𝑀))
28	3, 27	ringcl 19868	. . . . . . . . . . . . . 14 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ 𝐶 ∈ 𝑅 ∧ (𝑥‘𝑣) ∈ 𝑅) → (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)) ∈ 𝑅)
29	17, 19, 26, 28	syl3anc 1370	. . . . . . . . . . . . 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 7028	. . . . . . . . . . . 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 6825	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ V
32		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
33	32	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → 𝑉 ∈ 𝒫 (Base‘𝑀))
34	33	adantl 482	. . . . . . . . . . . . 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 8676	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 256	. . . . . . . . . . 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 1127	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶 ∈ 𝑅))
39	38	adantl 482	. . . . . . . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ (𝑅 ↑m 𝑉))
41	40	ad2antrr 723	. . . . . . . . . . . 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 768	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑m 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
43	42	ad2antrr 723	. . . . . . . . . . . 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 45962	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 7323	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
47	46	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
48	47	adantl 482	. . . . . . . . . . . . 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 723	. . . . . . . . . . . 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 768	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 613	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
54		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣))) = (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘(Scalar‘𝑀))(𝑥‘𝑣)))
55	11, 27, 53, 3, 54	lincscm 46023	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 2777	. . . . . . . . . . 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 5090	. . . . . . . . . . . . 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 7322	. . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 3570	. . . . . . . . . . 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 834	. . . . . . . . . 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 413	. . . . . . . . 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 413	. . . . . . . 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 3141	. . . . . . 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 408	. . . . . 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 408	. . . . 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 46005	. . . . . 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 723	. . . . 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 710	. . . 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 413	. . 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 239	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉)))
74	73	3impia 1116	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∃wrex 3071  Vcvv 3441  𝒫 cpw 4545   class class class wbr 5087   ↦ cmpt 5170  ⟶wf 6461  ‘cfv 6465  (class class class)co 7315   ↑_m cmap 8663   finSupp cfsupp 9198  Basecbs 16982  ._rcmulr 17033  Scalarcsca 17035   ·_𝑠 cvsca 17036  0_gc0g 17220  Ringcrg 19851  LModclmod 20195   linC clinc 45997   LinCo clinco 45998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628  ax-cnex 11000  ax-resscn 11001  ax-1cn 11002  ax-icn 11003  ax-addcl 11004  ax-addrcl 11005  ax-mulcl 11006  ax-mulrcl 11007  ax-mulcom 11008  ax-addass 11009  ax-mulass 11010  ax-distr 11011  ax-i2m1 11012  ax-1ne0 11013  ax-1rid 11014  ax-rnegex 11015  ax-rrecex 11016  ax-cnre 11017  ax-pre-lttri 11018  ax-pre-lttrn 11019  ax-pre-ltadd 11020  ax-pre-mulgt0 11021

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5562  df-se 5563  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-isom 6474  df-riota 7272  df-ov 7318  df-oprab 7319  df-mpo 7320  df-om 7758  df-1st 7876  df-2nd 7877  df-supp 8025  df-frecs 8144  df-wrecs 8175  df-recs 8249  df-rdg 8288  df-1o 8344  df-er 8546  df-map 8665  df-en 8782  df-dom 8783  df-sdom 8784  df-fin 8785  df-fsupp 9199  df-oi 9339  df-card 9768  df-pnf 11084  df-mnf 11085  df-xr 11086  df-ltxr 11087  df-le 11088  df-sub 11280  df-neg 11281  df-nn 12047  df-2 12109  df-n0 12307  df-z 12393  df-uz 12656  df-fz 13313  df-fzo 13456  df-seq 13795  df-hash 14118  df-sets 16935  df-slot 16953  df-ndx 16965  df-base 16983  df-plusg 17045  df-0g 17222  df-gsum 17223  df-mgm 18396  df-sgrp 18445  df-mnd 18456  df-mhm 18500  df-grp 18649  df-minusg 18650  df-ghm 18901  df-cntz 18992  df-cmn 19456  df-abl 19457  df-mgp 19789  df-ur 19806  df-ring 19853  df-lmod 20197  df-linc 45999  df-lco 46000

This theorem is referenced by:  lincsumscmcl  46026