lincsumcl - Mathbox for Alexander van der Vekens

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Theorem lincsumcl 43085

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 2825	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2825	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2825	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
4	1, 2, 3	lcoval 43066	. . . 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 43066	. . . 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 624	. . 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 783	. . . . . 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 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)))) → 𝐶 ∈ (Base‘𝑀))
9	8	adantl 475	. . . . . 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 787	. . . . . . 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 475	. . . . . 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 19240	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 + 𝐷) ∈ (Base‘𝑀))
14	7, 9, 11, 13	syl3anc 1494	. . . . 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 19235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Grp)
16		grpmnd 17790	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Scalar‘𝑀) ∈ Grp → (Scalar‘𝑀) ∈ Mnd)
17	15, 16	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Mnd)
18	17	adantr 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Mnd)
19	18	adantl 475	. . . . . . . . . . . . . . . 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)
20		simpr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
21	20	adantl 475	. . . . . . . . . . . . . . . 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))
22		simpll 783	. . . . . . . . . . . . . . . . . 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))
23		simpl 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_𝑚 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_𝑚 𝑉))
24	22, 23	anim12i 606	. . . . . . . . . . . . . . . . 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)))
25	24	adantr 474	. . . . . . . . . . . . . . . 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)))
26		eqid 2825	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘(Scalar‘𝑀)) = (+_g‘(Scalar‘𝑀))
27	3, 26	ofaddmndmap 42987	. . . . . . . . . . . . . . . 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))
28	19, 21, 25, 27	syl3anc 1494	. . . . . . . . . . . . . . 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))
29	17	anim1i 608	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
30	29	adantl 475	. . . . . . . . . . . . . . . 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)))
31		simprl 787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_𝑚 𝑉) ∧ (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → 𝑦 finSupp (0_g‘(Scalar‘𝑀)))
32	31	adantr 474	. . . . . . . . . . . . . . . . . 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)))
33		simprl 787	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_𝑚 𝑉) ∧ (𝑥 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0_g‘(Scalar‘𝑀)))
34	32, 33	anim12i 606	. . . . . . . . . . . . . . . . 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))))
35	34	adantr 474	. . . . . . . . . . . . . . . 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))))
36	3	mndpfsupp 43022	. . . . . . . . . . . . . . . 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)))
37	30, 25, 35, 36	syl3anc 1494	. . . . . . . . . . . . . . 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)))
38		oveq12 6919	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐶 = (𝑦( linC ‘𝑀)𝑉) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
39	38	expcom 404	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
40	39	adantl 475	. . . . . . . . . . . . . . . . . . . . . . 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))))
41	40	adantl 475	. . . . . . . . . . . . . . . . . . . . . 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))))
42	41	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))))
43	42	adantl 475	. . . . . . . . . . . . . . . . . . . 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))))
44	43	adantl 475	. . . . . . . . . . . . . . . . . . 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))))
45	44	adantr 474	. . . . . . . . . . . . . . . . . 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))))
46	45	imp 397	. . . . . . . . . . . . . . . . 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)))
47	46	adantr 474	. . . . . . . . . . . . . . . 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)))
48		simpr 479	. . . . . . . . . . . . . . . . 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)))
49		eqid 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦( linC ‘𝑀)𝑉) = (𝑦( linC ‘𝑀)𝑉)
50		eqid 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
51	12, 49, 50, 2, 3, 26	lincsum 43083	. . . . . . . . . . . . . . . . 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))
52	48, 25, 35, 51	syl3anc 1494	. . . . . . . . . . . . . . . 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))
53	47, 52	eqtrd 2861	. . . . . . . . . . . . . . 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))
54		breq1 4878	. . . . . . . . . . . . . . . . 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))))
55		oveq1 6917	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑦 ∘_𝑓 (+_g‘(Scalar‘𝑀))𝑥) → (𝑠( linC ‘𝑀)𝑉) = ((𝑦 ∘_𝑓 (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
56	55	eqeq2d 2835	. . . . . . . . . . . . . . . . 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)))
57	54, 56	anbi12d 624	. . . . . . . . . . . . . . . 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))))
58	57	rspcev 3526	. . . . . . . . . . . . . . 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)))
59	28, 37, 53, 58	syl12anc 870	. . . . . . . . . . . . . 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)))
60	59	exp41 427	. . . . . . . . . . . . 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))))))
61	60	rexlimiva 3237	. . . . . . . . . . . 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))))))
62	61	expd 406	. . . . . . . . . . 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)))))))
63	62	impcom 398	. . . . . . . . . 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))))))
64	63	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))))))
65	64	rexlimiva 3237	. . . . . . . 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))))))
66	65	impcom 398	. . . . . . 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)))))
67	66	impcom 398	. . . . . 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))))
68	67	impcom 398	. . . . 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)))
69	1, 2, 3	lcoval 43066	. . . . . 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)))))
70	69	adantr 474	. . . . 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)))))
71	14, 68, 70	mpbir2and 704	. . . 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))
72	71	ex 403	. . 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)))
73	6, 72	sylbid 232	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
74	73	imp 397	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 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 𝒫 cpw 4380 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ∘_𝑓 cof 7160 ↑_𝑚 cmap 8127 finSupp cfsupp 8550 Basecbs 16229 +_gcplusg 16312 Scalarcsca 16315 0_gc0g 16460 Mndcmnd 17654 Grpcgrp 17783 LModclmod 19226 linC clinc 43058 LinCo clinco 43059

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-seq 13103 df-hash 13418 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-0g 16462 df-gsum 16463 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-cntz 18107 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-lmod 19228 df-linc 43060 df-lco 43061

This theorem is referenced by: lincsumscmcl 43087

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