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Theorem lincsumcl 42011
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 + = (+g𝑀)
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.
StepHypRef Expression
1 eqid 2604 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2604 . . . . 5 (Scalar‘𝑀) = (Scalar‘𝑀)
3 eqid 2604 . . . . 5 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
41, 2, 3lcoval 41992 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉)))))
51, 2, 3lcoval 41992 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))))
64, 5anbi12d 742 . . 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 (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))))
7 simpll 785 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → 𝑀 ∈ LMod)
8 simpll 785 . . . . . . 7 (((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐶 ∈ (Base‘𝑀))
98adantl 480 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → 𝐶 ∈ (Base‘𝑀))
10 simprl 789 . . . . . . 7 (((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → 𝐷 ∈ (Base‘𝑀))
1110adantl 480 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → 𝐷 ∈ (Base‘𝑀))
12 lincsumcl.b . . . . . . 7 + = (+g𝑀)
131, 12lmodvacl 18641 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → (𝐶 + 𝐷) ∈ (Base‘𝑀))
147, 9, 11, 13syl3anc 1317 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → (𝐶 + 𝐷) ∈ (Base‘𝑀))
152lmodfgrp 18636 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Grp)
16 grpmnd 17193 . . . . . . . . . . . . . . . . . . 19 ((Scalar‘𝑀) ∈ Grp → (Scalar‘𝑀) ∈ Mnd)
1715, 16syl 17 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ LMod → (Scalar‘𝑀) ∈ Mnd)
1817adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (Scalar‘𝑀) ∈ Mnd)
1918adantl 480 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (Scalar‘𝑀) ∈ Mnd)
20 simpr 475 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ∈ 𝒫 (Base‘𝑀))
2120adantl 480 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → 𝑉 ∈ 𝒫 (Base‘𝑀))
22 simpll 785 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) → 𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
23 simpl 471 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
2422, 23anim12i 587 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)))
2524adantr 479 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(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 2604 . . . . . . . . . . . . . . . . 17 (+g‘(Scalar‘𝑀)) = (+g‘(Scalar‘𝑀))
273, 26ofaddmndmap 41912 . . . . . . . . . . . . . . . 16 (((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))) → (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
2819, 21, 25, 27syl3anc 1317 . . . . . . . . . . . . . . 15 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
2917anim1i 589 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
3029adantl 480 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
31 simprl 789 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → 𝑦 finSupp (0g‘(Scalar‘𝑀)))
3231adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) → 𝑦 finSupp (0g‘(Scalar‘𝑀)))
33 simprl 789 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → 𝑥 finSupp (0g‘(Scalar‘𝑀)))
3432, 33anim12i 587 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0g‘(Scalar‘𝑀))))
3534adantr 479 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0g‘(Scalar‘𝑀))))
363mndpfsupp 41948 . . . . . . . . . . . . . . . 16 ((((Scalar‘𝑀) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0g‘(Scalar‘𝑀)))) → (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) finSupp (0g‘(Scalar‘𝑀)))
3730, 25, 35, 36syl3anc 1317 . . . . . . . . . . . . . . 15 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) finSupp (0g‘(Scalar‘𝑀)))
38 oveq12 6531 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐶 = (𝑦( linC ‘𝑀)𝑉) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
3938expcom 449 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐷 = (𝑥( linC ‘𝑀)𝑉) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
4039adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
4140adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 = (𝑦( linC ‘𝑀)𝑉) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
4241com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐶 = (𝑦( linC ‘𝑀)𝑉) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
4342adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉)) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
4443adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
4544adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉))))
4645imp 443 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 + 𝐷) = ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)))
4746adantr 479 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(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 475 . . . . . . . . . . . . . . . . 17 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))
49 eqid 2604 . . . . . . . . . . . . . . . . . 18 (𝑦( linC ‘𝑀)𝑉) = (𝑦( linC ‘𝑀)𝑉)
50 eqid 2604 . . . . . . . . . . . . . . . . . 18 (𝑥( linC ‘𝑀)𝑉) = (𝑥( linC ‘𝑀)𝑉)
5112, 49, 50, 2, 3, 26lincsum 42009 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑥 finSupp (0g‘(Scalar‘𝑀)))) → ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)) = ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
5248, 25, 35, 51syl3anc 1317 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ((𝑦( linC ‘𝑀)𝑉) + (𝑥( linC ‘𝑀)𝑉)) = ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
5347, 52eqtrd 2638 . . . . . . . . . . . . . . 15 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝐶 + 𝐷) = ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
54 breq1 4575 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) finSupp (0g‘(Scalar‘𝑀))))
55 oveq1 6529 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) → (𝑠( linC ‘𝑀)𝑉) = ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))
5655eqeq2d 2614 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) → ((𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉) ↔ (𝐶 + 𝐷) = ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉)))
5754, 56anbi12d 742 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) → ((𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))))
5857rspcev 3276 . . . . . . . . . . . . . . 15 (((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥) finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = ((𝑦𝑓 (+g‘(Scalar‘𝑀))𝑥)( linC ‘𝑀)𝑉))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
5928, 37, 53, 58syl12anc 1315 . . . . . . . . . . . . . 14 (((((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀))) ∧ (𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
6059exp41 635 . . . . . . . . . . . . 13 ((𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
6160rexlimiva 3004 . . . . . . . . . . . 12 (∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉)) → ((𝐶 ∈ (Base‘𝑀) ∧ 𝐷 ∈ (Base‘𝑀)) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
6261expd 450 . . . . . . . . . . 11 (∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉)) → (𝐶 ∈ (Base‘𝑀) → (𝐷 ∈ (Base‘𝑀) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))))
6362impcom 444 . . . . . . . . . 10 ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → (𝐷 ∈ (Base‘𝑀) → ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
6463com13 85 . . . . . . . . 9 ((𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → (𝐷 ∈ (Base‘𝑀) → ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
6564rexlimiva 3004 . . . . . . . 8 (∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)) → (𝐷 ∈ (Base‘𝑀) → ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))))
6665impcom 444 . . . . . . 7 ((𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))) → ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
6766impcom 444 . . . . . 6 (((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉))))
6867impcom 444 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))
691, 2, 3lcoval 41992 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 + 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
7069adantr 479 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → ((𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐶 + 𝐷) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (𝐶 + 𝐷) = (𝑠( linC ‘𝑀)𝑉)))))
7114, 68, 70mpbir2and 958 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ ((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉))))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉))
7271ex 448 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (((𝐶 ∈ (Base‘𝑀) ∧ ∃𝑦 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑦 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐶 = (𝑦( linC ‘𝑀)𝑉))) ∧ (𝐷 ∈ (Base‘𝑀) ∧ ∃𝑥 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑥 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝐷 = (𝑥( linC ‘𝑀)𝑉)))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
736, 72sylbid 228 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
7473imp 443 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 194  wa 382   = wceq 1474  wcel 1975  wrex 2891  𝒫 cpw 4102   class class class wbr 4572  cfv 5785  (class class class)co 6522  𝑓 cof 6765  𝑚 cmap 7716   finSupp cfsupp 8130  Basecbs 15636  +gcplusg 15709  Scalarcsca 15712  0gc0g 15864  Mndcmnd 17058  Grpcgrp 17186  LModclmod 18627   linC clinc 41984   LinCo clinco 41985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-cnex 9843  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-pre-mulgt0 9864
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-of 6767  df-om 6930  df-1st 7031  df-2nd 7032  df-supp 7155  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-1o 7419  df-oadd 7423  df-er 7601  df-map 7718  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-fsupp 8131  df-oi 8270  df-card 8620  df-pnf 9927  df-mnf 9928  df-xr 9929  df-ltxr 9930  df-le 9931  df-sub 10114  df-neg 10115  df-nn 10863  df-2 10921  df-n0 11135  df-z 11206  df-uz 11515  df-fz 12148  df-fzo 12285  df-seq 12614  df-hash 12930  df-ndx 15639  df-slot 15640  df-base 15641  df-sets 15642  df-ress 15643  df-plusg 15722  df-0g 15866  df-gsum 15867  df-mgm 17006  df-sgrp 17048  df-mnd 17059  df-submnd 17100  df-grp 17189  df-minusg 17190  df-cntz 17514  df-cmn 17959  df-abl 17960  df-mgp 18254  df-ur 18266  df-ring 18313  df-lmod 18629  df-linc 41986  df-lco 41987
This theorem is referenced by:  lincsumscmcl  42013
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