lincscmcl - Mathbox for Alexander van der Vekens

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Theorem lincscmcl 47113

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 2733	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2733	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		lincscmcl.r	. . . . 5 ⊢ 𝑅 = (Base‘(Scalar‘𝑀))
4	1, 2, 3	lcoval 47093	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))))
5	4	adantr 482	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))))
6		simpl 484	. . . . . . 7 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑀 ∈ LMod)
7	6	ad2antrr 725	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝑀 ∈ LMod)
8		simpr 486	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → 𝐶 ∈ 𝑅)
9	8	adantr 482	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐶 ∈ 𝑅)
10		simprl 770	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐷 ∈ (Base‘𝑀))
11		lincscmcl.s	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑀)
12	1, 2, 11, 3	lmodvscl 20489	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
13	7, 9, 10, 12	syl3anc 1372	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 · 𝐷) ∈ (Base‘𝑀))
14	2	lmodring 20479	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Ring)
15	14	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → (Scalar‘𝑀) ∈ Ring)
16	15	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → (Scalar‘𝑀) ∈ Ring)
17	16	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → 𝐶 ∈ 𝑅)
19	18	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ 𝑅)
20		elmapi 8843	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝑅 ↑_m 𝑉) → 𝑥:𝑉⟶𝑅)
21		ffvelcdm 7084	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥:𝑉⟶𝑅 ∧ 𝑣 ∈ 𝑉) → (𝑥‘𝑣) ∈ 𝑅)
22	21	ex 414	. . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝑣 ∈ 𝑉 → (𝑥‘𝑣) ∈ 𝑅))
25	24	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 408	. . . . . . . . . . . . . 14 ⊢ (((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 2733	. . . . . . . . . . . . . . 15 ⊢ (._r‘(Scalar‘𝑀)) = (._r‘(Scalar‘𝑀))
28	3, 27	ringcl 20073	. . . . . . . . . . . . . 14 ⊢ (((Scalar‘𝑀) ∈ Ring ∧ 𝐶 ∈ 𝑅 ∧ (𝑥‘𝑣) ∈ 𝑅) → (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣)) ∈ 𝑅)
29	17, 19, 26, 28	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ (((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 7115	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 6906	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ V
32		simpr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
33	32	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → 𝑉 ∈ 𝒫 (Base‘𝑀))
34	33	adantl 483	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
35		elmapg 8833	. . . . . . . . . . . . 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 588	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 257	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 1129	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶 ∈ 𝑅))
39	38	adantl 483	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 484	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ (𝑅 ↑_m 𝑉))
41	40	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → 𝑥 ∈ (𝑅 ↑_m 𝑉))
42		simprl 770	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0_g‘(Scalar‘𝑀)))
43	42	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → 𝑥 finSupp (0_g‘(Scalar‘𝑀)))
44	3	rmfsupp 47050	. . . . . . . . . . . 12 ⊢ ((((Scalar‘𝑀) ∈ Ring ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐶 ∈ 𝑅) ∧ 𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ 𝑥 finSupp (0_g‘(Scalar‘𝑀))) → (𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0_g‘(Scalar‘𝑀)))
45	39, 41, 43, 44	syl3anc 1372	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 (0_g‘(Scalar‘𝑀)))
46		oveq2 7417	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
47	46	adantl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
48	47	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 · 𝐷) = (𝐶 · (𝑥( linC ‘𝑀)𝑉)))
49	48	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 770	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 482	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → 𝑥 ∈ (𝑅 ↑_m 𝑉))
52	51, 8	anim12i 614	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 2733	. . . . . . . . . . . . . 14 ⊢ (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
54		eqid 2733	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣))) = (𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣)))
55	11, 27, 53, 3, 54	lincscm 47111	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ 𝐶 ∈ 𝑅) ∧ 𝑥 finSupp (0_g‘(Scalar‘𝑀))) → (𝐶 · (𝑥( linC ‘𝑀)𝑉)) = ((𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))
56	50, 52, 43, 55	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 2773	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(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 5152	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣))) → (𝑠 finSupp (0_g‘(Scalar‘𝑀)) ↔ (𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0_g‘(Scalar‘𝑀))))
59		oveq1 7416	. . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . 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 632	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣))) → ((𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))))
62	61	rspcev 3613	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣))) ∈ (𝑅 ↑_m 𝑉) ∧ ((𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣))) finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = ((𝑣 ∈ 𝑉 ↦ (𝐶(._r‘(Scalar‘𝑀))(𝑥‘𝑣)))( linC ‘𝑀)𝑉))) → ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
63	37, 45, 57, 62	syl12anc 836	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) ∧ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅)) → ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
64	63	ex 414	. . . . . . . . 9 ⊢ (((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) ∧ 𝐷 ∈ (Base‘𝑀)) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉))))
65	64	ex 414	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝑅 ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐷 ∈ (Base‘𝑀) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
66	65	rexlimiva 3148	. . . . . . 7 ⊢ (∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐷 ∈ (Base‘𝑀) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
67	66	impcom 409	. . . . . 6 ⊢ ((𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉))))
68	67	impcom 409	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
69	1, 2, 3	lcoval 47093	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 · 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
70	69	ad2antrr 725	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → ((𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 · 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ (𝑅 ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 · 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
71	13, 68, 70	mpbir2and 712	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))
72	71	ex 414	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅) → ((𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ (𝑅 ↑_m 𝑉)(𝑥 finSupp (0_g‘(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 1118	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶 ∈ 𝑅 ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 Vcvv 3475 𝒫 cpw 4603 class class class wbr 5149 ↦ cmpt 5232 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 finSupp cfsupp 9361 Basecbs 17144 ._rcmulr 17198 Scalarcsca 17200 ·_𝑠 cvsca 17201 0_gc0g 17385 Ringcrg 20056 LModclmod 20471 linC clinc 47085 LinCo clinco 47086

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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-0g 17387 df-gsum 17388 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-grp 18822 df-minusg 18823 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-lmod 20473 df-linc 47087 df-lco 47088

This theorem is referenced by: lincsumscmcl 47114

W3C validator