lincsumcl - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lincsumcl		Structured version Visualization version GIF version

Theorem lincsumcl 47814

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 2726	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2726	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2726	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
4	1, 2, 3	lcoval 47795	. . . 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 47795	. . . 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 630	. . 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 765	. . . . . 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 765	. . . . . . 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 480	. . . . . 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 769	. . . . . . 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 480	. . . . . 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 20851	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 + 𝐷) ∈ (Base‘𝑀))
14	7, 9, 11, 13	syl3anc 1368	. . . . 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 20845	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Grp)
16	15	grpmndd 18941	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Mnd)
17	16	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Mnd)
18	17	adantl 480	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
20	19	adantl 480	. . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 611	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘(Scalar‘𝑀)) = (+_g‘(Scalar‘𝑀))
26	3, 25	ofaddmndmap 47722	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 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 613	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
29	28	adantl 480	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . 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 611	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 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 47755	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐶 = (𝑦( linC ‘𝑀)𝑉) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
38	37	expcom 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
39	38	adantl 480	. . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦( linC ‘𝑀)𝑉) = (𝑦( linC ‘𝑀)𝑉)
49		eqid 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
50	12, 48, 49, 2, 3, 25	lincsum 47812	. . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 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 5156	. . . . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥) → (𝑠( linC ‘𝑀)𝑉) = ((𝑦 ∘_f (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
55	54	eqeq2d 2737	. . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . 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 3608	. . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . 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 433	. . . . . . . . . . . . 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 3137	. . . . . . . . . . . 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 414	. . . . . . . . . . 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 406	. . . . . . . . . 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 88	. . . . . . . . 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 3137	. . . . . . . 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 406	. . . . . . 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 406	. . . . . 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 406	. . . . 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 47795	. . . . . 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 479	. . . . 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 711	. . . 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 411	. . 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 239	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
73	72	imp 405	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

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃wrex 3060 𝒫 cpw 4607 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 ∘_f cof 7688 ↑_m cmap 8855 finSupp cfsupp 9405 Basecbs 17213 +_gcplusg 17266 Scalarcsca 17269 0_gc0g 17454 Mndcmnd 18727 LModclmod 20836 linC clinc 47787 LinCo clinco 47788

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-0g 17456 df-gsum 17457 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-ur 20165 df-ring 20218 df-lmod 20838 df-linc 47789 df-lco 47790

This theorem is referenced by: lincsumscmcl 47816

W3C validator