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 42014

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 2610	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2610	. . . . 5 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
3		eqid 2610	. . . . 5 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
4	1, 2, 3	lcoval 41995	. . . 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 41995	. . . 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 743	. . 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 786	. . . . . 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 786	. . . . . . 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 481	. . . . . 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 790	. . . . . . 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 481	. . . . . 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 18700	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 + 𝐷) ∈ (Base‘𝑀))
14	7, 9, 11, 13	syl3anc 1318	. . . . 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 18695	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Grp)
16		grpmnd 17252	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Scalar‘𝑀) ∈ Grp → (Scalar‘𝑀) ∈ Mnd)
17	15, 16	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Mnd)
18	17	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Mnd)
19	18	adantl 481	. . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
21	20	adantl 481	. . . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . . . 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 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))) → (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑_𝑚 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑_𝑚 𝑉)))
26		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘(Scalar‘𝑀)) = (+_g‘(Scalar‘𝑀))
27	3, 26	ofaddmndmap 41915	. . . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
30	29	adantl 481	. . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . . . 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 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))) → (𝑦 finSupp (0_g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0_g‘(Scalar‘𝑀))))
36	3	mndpfsupp 41951	. . . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . . 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 6558	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐶 = (𝑦( linC ‘𝑀)𝑉) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
39	38	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
40	39	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . . 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 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))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
48		simpr 476	. . . . . . . . . . . . . . . . 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 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦( linC ‘𝑀)𝑉) = (𝑦( linC ‘𝑀)𝑉)
50		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
51	12, 49, 50, 2, 3, 26	lincsum 42012	. . . . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . 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 4586	. . . . . . . . . . . . . . . . 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 6556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = (𝑦 ∘_𝑓 (+_g‘(Scalar‘𝑀))𝑥) → (𝑠( linC ‘𝑀)𝑉) = ((𝑦 ∘_𝑓 (+_g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
56	55	eqeq2d 2620	. . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . 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 3282	. . . . . . . . . . . . . . 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 1316	. . . . . . . . . . . . . 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 636	. . . . . . . . . . . . 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 3010	. . . . . . . . . . . 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 451	. . . . . . . . . . 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 445	. . . . . . . . . 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 86	. . . . . . . . 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 3010	. . . . . . . 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 445	. . . . . . 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 445	. . . . . 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 445	. . . . 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 41995	. . . . . 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 480	. . . . 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 959	. . . 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 449	. . 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 229	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
74	73	imp 444	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 𝒫 cpw 4108 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ↑_𝑚 cmap 7744 finSupp cfsupp 8158 Basecbs 15695 +_gcplusg 15768 Scalarcsca 15771 0_gc0g 15923 Mndcmnd 17117 Grpcgrp 17245 LModclmod 18686 linC clinc 41987 LinCo clinco 41988

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-linc 41989 df-lco 41990

This theorem is referenced by: lincsumscmcl 42016

W3C validator