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Theorem lincsumcl 47611
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 + = (+gβ€˜π‘€)
Assertion
Ref Expression
lincsumcl (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐢 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) β†’ (𝐢 + 𝐷) ∈ (𝑀 LinCo 𝑉))

Proof of Theorem lincsumcl
Dummy variables 𝑠 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . . . . 5 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2725 . . . . 5 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
3 eqid 2725 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
41, 2, 3lcoval 47592 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝐢 ∈ (𝑀 LinCo 𝑉) ↔ (𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉)))))
51, 2, 3lcoval 47592 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝐷 ∈ (𝑀 LinCo 𝑉) ↔ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))))
64, 5anbi12d 630 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝐢 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) ↔ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))))))
7 simpll 765 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))))) β†’ 𝑀 ∈ LMod)
8 simpll 765 . . . . . . 7 (((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ 𝐢 ∈ (Baseβ€˜π‘€))
98adantl 480 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))))) β†’ 𝐢 ∈ (Baseβ€˜π‘€))
10 simprl 769 . . . . . . 7 (((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ 𝐷 ∈ (Baseβ€˜π‘€))
1110adantl 480 . . . . . 6 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))))) β†’ 𝐷 ∈ (Baseβ€˜π‘€))
12 lincsumcl.b . . . . . . 7 + = (+gβ€˜π‘€)
131, 12lmodvacl 20762 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) β†’ (𝐢 + 𝐷) ∈ (Baseβ€˜π‘€))
147, 9, 11, 13syl3anc 1368 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))))) β†’ (𝐢 + 𝐷) ∈ (Baseβ€˜π‘€))
152lmodfgrp 20756 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) ∈ Grp)
1615grpmndd 18907 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) ∈ Mnd)
1716adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (Scalarβ€˜π‘€) ∈ Mnd)
1817adantl 480 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (Scalarβ€˜π‘€) ∈ Mnd)
19 simpr 483 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
2019adantl 480 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
21 simpll 765 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) β†’ 𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
22 simpl 481 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
2321, 22anim12i 611 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ (𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)))
2423adantr 479 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)))
25 eqid 2725 . . . . . . . . . . . . . . . . 17 (+gβ€˜(Scalarβ€˜π‘€)) = (+gβ€˜(Scalarβ€˜π‘€))
263, 25ofaddmndmap 47519 . . . . . . . . . . . . . . . 16 (((Scalarβ€˜π‘€) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ (𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))) β†’ (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
2718, 20, 24, 26syl3anc 1368 . . . . . . . . . . . . . . 15 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
2816anim1i 613 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((Scalarβ€˜π‘€) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))
2928adantl 480 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ ((Scalarβ€˜π‘€) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))
30 simprl 769 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) β†’ 𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)))
3130adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) β†’ 𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)))
32 simprl 769 . . . . . . . . . . . . . . . . . 18 ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)))
3331, 32anim12i 611 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€))))
3433adantr 479 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€))))
353mndpfsupp 47552 . . . . . . . . . . . . . . . 16 ((((Scalarβ€˜π‘€) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)))) β†’ (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) finSupp (0gβ€˜(Scalarβ€˜π‘€)))
3629, 24, 34, 35syl3anc 1368 . . . . . . . . . . . . . . 15 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) finSupp (0gβ€˜(Scalarβ€˜π‘€)))
37 oveq12 7425 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐢 = (𝑦( linC β€˜π‘€)𝑉) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉)))
3837expcom 412 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐷 = (π‘₯( linC β€˜π‘€)𝑉) β†’ (𝐢 = (𝑦( linC β€˜π‘€)𝑉) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉))))
3938adantl 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)) β†’ (𝐢 = (𝑦( linC β€˜π‘€)𝑉) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉))))
4039adantl 480 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐢 = (𝑦( linC β€˜π‘€)𝑉) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉))))
4140com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝐢 = (𝑦( linC β€˜π‘€)𝑉) β†’ ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉))))
4241adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉)) β†’ ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉))))
4342adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) β†’ ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉))))
4443adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) β†’ ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉))))
4544imp 405 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉)))
4645adantr 479 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉)))
47 simpr 483 . . . . . . . . . . . . . . . . 17 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))
48 eqid 2725 . . . . . . . . . . . . . . . . . 18 (𝑦( linC β€˜π‘€)𝑉) = (𝑦( linC β€˜π‘€)𝑉)
49 eqid 2725 . . . . . . . . . . . . . . . . . 18 (π‘₯( linC β€˜π‘€)𝑉) = (π‘₯( linC β€˜π‘€)𝑉)
5012, 48, 49, 2, 3, 25lincsum 47609 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)))) β†’ ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉)) = ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯)( linC β€˜π‘€)𝑉))
5147, 24, 34, 50syl3anc 1368 . . . . . . . . . . . . . . . 16 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉)) = ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯)( linC β€˜π‘€)𝑉))
5246, 51eqtrd 2765 . . . . . . . . . . . . . . 15 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (𝐢 + 𝐷) = ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯)( linC β€˜π‘€)𝑉))
53 breq1 5146 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) β†’ (𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ↔ (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) finSupp (0gβ€˜(Scalarβ€˜π‘€))))
54 oveq1 7423 . . . . . . . . . . . . . . . . . 18 (𝑠 = (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) β†’ (𝑠( linC β€˜π‘€)𝑉) = ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯)( linC β€˜π‘€)𝑉))
5554eqeq2d 2736 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) β†’ ((𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉) ↔ (𝐢 + 𝐷) = ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯)( linC β€˜π‘€)𝑉)))
5653, 55anbi12d 630 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) β†’ ((𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉)) ↔ ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯)( linC β€˜π‘€)𝑉))))
5756rspcev 3601 . . . . . . . . . . . . . . 15 (((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = ((𝑦 ∘f (+gβ€˜(Scalarβ€˜π‘€))π‘₯)( linC β€˜π‘€)𝑉))) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))
5827, 36, 52, 57syl12anc 835 . . . . . . . . . . . . . 14 (((((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€))) ∧ (π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))
5958exp41 433 . . . . . . . . . . . . 13 ((𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) β†’ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) β†’ ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉))))))
6059rexlimiva 3137 . . . . . . . . . . . 12 (βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉)) β†’ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) β†’ ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉))))))
6160expd 414 . . . . . . . . . . 11 (βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉)) β†’ (𝐢 ∈ (Baseβ€˜π‘€) β†’ (𝐷 ∈ (Baseβ€˜π‘€) β†’ ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))))))
6261impcom 406 . . . . . . . . . 10 ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) β†’ (𝐷 ∈ (Baseβ€˜π‘€) β†’ ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉))))))
6362com13 88 . . . . . . . . 9 ((π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ (𝐷 ∈ (Baseβ€˜π‘€) β†’ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) β†’ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉))))))
6463rexlimiva 3137 . . . . . . . 8 (βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)) β†’ (𝐷 ∈ (Baseβ€˜π‘€) β†’ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) β†’ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉))))))
6564impcom 406 . . . . . . 7 ((𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))) β†’ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) β†’ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))))
6665impcom 406 . . . . . 6 (((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉))))
6766impcom 406 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))))) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))
681, 2, 3lcoval 47592 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝐢 + 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐢 + 𝐷) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))))
6968adantr 479 . . . . 5 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))))) β†’ ((𝐢 + 𝐷) ∈ (𝑀 LinCo 𝑉) ↔ ((𝐢 + 𝐷) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (𝐢 + 𝐷) = (𝑠( linC β€˜π‘€)𝑉)))))
7014, 67, 69mpbir2and 711 . . . 4 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ ((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉))))) β†’ (𝐢 + 𝐷) ∈ (𝑀 LinCo 𝑉))
7170ex 411 . . 3 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (((𝐢 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘¦ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐢 = (𝑦( linC β€˜π‘€)𝑉))) ∧ (𝐷 ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ 𝐷 = (π‘₯( linC β€˜π‘€)𝑉)))) β†’ (𝐢 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
726, 71sylbid 239 . 2 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝐢 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉)) β†’ (𝐢 + 𝐷) ∈ (𝑀 LinCo 𝑉)))
7372imp 405 1 (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ (𝐢 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) β†’ (𝐢 + 𝐷) ∈ (𝑀 LinCo 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060  π’« cpw 4598   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416   ∘f cof 7680   ↑m cmap 8843   finSupp cfsupp 9385  Basecbs 17179  +gcplusg 17232  Scalarcsca 17235  0gc0g 17420  Mndcmnd 18693  LModclmod 20747   linC clinc 47584   LinCo clinco 47585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7991  df-2nd 7992  df-supp 8164  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-fsupp 9386  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-fzo 13660  df-seq 13999  df-hash 14322  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-0g 17422  df-gsum 17423  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18740  df-grp 18897  df-minusg 18898  df-cntz 19272  df-cmn 19741  df-abl 19742  df-mgp 20079  df-ur 20126  df-ring 20179  df-lmod 20749  df-linc 47586  df-lco 47587
This theorem is referenced by:  lincsumscmcl  47613
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