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Theorem lincsumcl 46598
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 2733 . . . . 5 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2733 . . . . 5 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
3 eqid 2733 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
41, 2, 3lcoval 46579 . . . 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 46579 . . . 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 632 . . 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 766 . . . . . 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 766 . . . . . . 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 483 . . . . . 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 770 . . . . . . 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 483 . . . . . 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 20351 . . . . . 6 ((𝑀 ∈ LMod ∧ 𝐢 ∈ (Baseβ€˜π‘€) ∧ 𝐷 ∈ (Baseβ€˜π‘€)) β†’ (𝐢 + 𝐷) ∈ (Baseβ€˜π‘€))
147, 9, 11, 13syl3anc 1372 . . . . 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 20345 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) ∈ Grp)
1615grpmndd 18765 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) ∈ Mnd)
1716adantr 482 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (Scalarβ€˜π‘€) ∈ Mnd)
1817adantl 483 . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
2019adantl 483 . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . 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 614 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . 17 (+gβ€˜(Scalarβ€˜π‘€)) = (+gβ€˜(Scalarβ€˜π‘€))
263, 25ofaddmndmap 46505 . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . 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 616 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((Scalarβ€˜π‘€) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))
2928adantl 483 . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . 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 614 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . 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 46538 . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . 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 7367 . . . . . . . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . . . 18 (𝑦( linC β€˜π‘€)𝑉) = (𝑦( linC β€˜π‘€)𝑉)
49 eqid 2733 . . . . . . . . . . . . . . . . . 18 (π‘₯( linC β€˜π‘€)𝑉) = (π‘₯( linC β€˜π‘€)𝑉)
5012, 48, 49, 2, 3, 25lincsum 46596 . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . 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 5109 . . . . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . . . . 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 2744 . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . 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 3580 . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . 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 3141 . . . . . . . . . . . 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 417 . . . . . . . . . . 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 409 . . . . . . . . . 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 3141 . . . . . . . 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 409 . . . . . . 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 409 . . . . . 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 409 . . . . 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 46579 . . . . . 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 482 . . . . 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 712 . . . 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 414 . . 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 408 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070  π’« cpw 4561   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358   ∘f cof 7616   ↑m cmap 8768   finSupp cfsupp 9308  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141  0gc0g 17326  Mndcmnd 18561  LModclmod 20336   linC clinc 46571   LinCo clinco 46572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-fzo 13574  df-seq 13913  df-hash 14237  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-0g 17328  df-gsum 17329  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-grp 18756  df-minusg 18757  df-cntz 19102  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-ring 19971  df-lmod 20338  df-linc 46573  df-lco 46574
This theorem is referenced by:  lincsumscmcl  46600
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