lincsumcl - Mathbox for Alexander van der Vekens

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Theorem lincsumcl 49270

Description: The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)

**Hypothesis**
Ref	Expression
lincsumcl.b	⊢ + = (+_g‘𝑀)

**Assertion**
Ref	Expression
lincsumcl	⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉))

Proof of Theorem lincsumcl Dummy variables 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2765	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2765	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
4	1, 2, 3	lcoval 49251	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉)))))
5	1, 2, 3	lcoval 49251	. . . 4 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))))
6	4, 5	anbi12d 644	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) ↔ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))))
7		simpll 779	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → 𝑀 ∈ LMod)
8		simpll 779	. . . . . . 7 ⊢ (((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐶 ∈ (Base‘𝑀))
9	8	adantl 487	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → 𝐶 ∈ (Base‘𝑀))
10		simprl 783	. . . . . . 7 ⊢ (((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐷 ∈ (Base‘𝑀))
11	10	adantl 487	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → 𝐷 ∈ (Base‘𝑀))
12		lincsumcl.b	. . . . . . 7 ⊢ + = (+_g‘𝑀)
13	1, 12	lmodvacl 21048	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 + 𝐷) ∈ (Base‘𝑀))
14	7, 9, 11, 13	syl3anc 1398	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → (𝐶 + 𝐷) ∈ (Base‘𝑀))
15	2	lmodfgrp 21042	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Grp)
16	15	grpmndd 19059	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Mnd)
17	16	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Mnd)
18	17	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (Scalar‘𝑀) ∈ Mnd)
19		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
20	19	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
21		simpll 779	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) → 𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
22		simpl 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
23	21, 22	anim12i 625	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)))
24	23	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)))
25		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘(Scalar‘𝑀)) = (+_g‘(Scalar‘𝑀))
26	3, 25	ofaddmndmap 49182	. . . . . . . . . . . . . . . 16 ⊢ (((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))) → (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
27	18, 20, 24, 26	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉))
28	16	anim1i 627	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
29	28	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
30		simprl 783	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → 𝑦 finSupp (0_g‘(Scalar‘𝑀)))
31	30	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) → 𝑦 finSupp (0_g‘(Scalar‘𝑀)))
32		simprl 783	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0_g‘(Scalar‘𝑀)))
33	31, 32	anim12i 625	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0_g‘(Scalar‘𝑀))))
34	33	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0_g‘(Scalar‘𝑀))))
35	3	mndpfsupp 18864	. . . . . . . . . . . . . . . 16 ⊢ ((((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0_g‘(Scalar‘𝑀)))) → (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) finSupp (0_g‘(Scalar‘𝑀)))
36	29, 24, 34, 35	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) finSupp (0_g‘(Scalar‘𝑀)))
37		oveq12 7428	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐶 = (𝑦( linC ‘𝑀)𝑉) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
38	37	expcom 419	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
39	38	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
40	39	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
41	40	com12 33	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐶 = (𝑦( linC ‘𝑀)𝑉) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
42	41	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉)) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
43	42	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
44	43	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
45	44	imp 412	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
46	45	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
47		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
48		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦( linC ‘𝑀)𝑉) = (𝑦( linC ‘𝑀)𝑉)
49		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
50	12, 48, 49, 2, 3, 25	lincsum 49268	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0_g‘(Scalar‘𝑀)))) → ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)) = ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
51	47, 24, 34, 50	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)) = ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
52	46, 51	eqtrd 2800	. . . . . . . . . . . . . . 15 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝐶 + 𝐷) = ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
53		breq1 5114	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) → (𝑠 finSupp (0_g‘(Scalar‘𝑀)) ↔ (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) finSupp (0_g‘(Scalar‘𝑀))))
54		oveq1 7426	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) → (𝑠( linC ‘𝑀)𝑉) = ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
55	54	eqeq2d 2776	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) → ((𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉) ↔ (𝐶 + 𝐷) = ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉)))
56	53, 55	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) → ((𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))))
57	56	rspcev 3583	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
58	27, 36, 52, 57	syl12anc 850	. . . . . . . . . . . . . 14 ⊢ (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
59	58	exp41 440	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
60	59	rexlimiva 3160	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉)) → ((𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
61	60	expd 421	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉)) → (𝐶 ∈ (Base‘𝑀) → (𝐷 ∈ (Base‘𝑀) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))))
62	61	impcom 413	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → (𝐷 ∈ (Base‘𝑀) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
63	62	com13 89	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐷 ∈ (Base‘𝑀) → ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
64	63	rexlimiva 3160	. . . . . . . 8 ⊢ (∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐷 ∈ (Base‘𝑀) → ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
65	64	impcom 413	. . . . . . 7 ⊢ ((𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
66	65	impcom 413	. . . . . 6 ⊢ (((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))
67	66	impcom 413	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
68	1, 2, 3	lcoval 49251	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 + 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
69	68	adantr 486	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → ((𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 + 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑠 finSupp (0_g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
70	14, 67, 69	mpbir2and 726	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉))
71	70	ex 418	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_m 𝑉)(𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
72	6, 71	sylbid 243	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
73	72	imp 412	1 ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∃wrex 3091 𝒫 cpw 4564 class class class wbr 5111 ‘cfv 6540 (class class class)co 7419 ∘_f cof 7682 ↑_m cmap 8830 finSupp cfsupp 9328 Basecbs 17293 +_gcplusg 17334 Scalarcsca 17337 0_gc0g 17516 Mndcmnd 18826 LModclmod 21033 linC clinc 49243 LinCo clinco 49244

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 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 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-n0 12522 df-z 12609 df-uz 12881 df-fz 13554 df-fzo 13702 df-seq 14058 df-hash 14387 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-0g 17518 df-gsum 17519 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-grp 19049 df-minusg 19050 df-cntz 19433 df-cmn 19898 df-abl 19899 df-mgp 20263 df-ur 20310 df-ring 20363 df-lmod 21035 df-linc 49245 df-lco 49246

This theorem is used by: lincsumscmcl 49272

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