Users' Mathboxes 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 47387
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 2726 . . . . 5 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
2 eqid 2726 . . . . 5 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
3 eqid 2726 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
41, 2, 3lcoval 47368 . . . 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 47368 . . . 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 764 . . . . . 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 764 . . . . . . 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 481 . . . . . 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 768 . . . . . . 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 481 . . . . . 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 20721 . . . . . 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 20715 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) ∈ Grp)
1615grpmndd 18876 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ LMod β†’ (Scalarβ€˜π‘€) ∈ Mnd)
1716adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (Scalarβ€˜π‘€) ∈ Mnd)
1817adantl 481 . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))
2019adantl 481 . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . . . 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 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†’ (𝑦 ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ π‘₯ ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)))
25 eqid 2726 . . . . . . . . . . . . . . . . 17 (+gβ€˜(Scalarβ€˜π‘€)) = (+gβ€˜(Scalarβ€˜π‘€))
263, 25ofaddmndmap 47295 . . . . . . . . . . . . . . . 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 614 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((Scalarβ€˜π‘€) ∈ Mnd ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))
2928adantl 481 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . . . 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 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†’ (𝑦 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ π‘₯ finSupp (0gβ€˜(Scalarβ€˜π‘€))))
353mndpfsupp 47328 . . . . . . . . . . . . . . . 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 7414 . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . 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 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†’ (𝐢 + 𝐷) = ((𝑦( linC β€˜π‘€)𝑉) + (π‘₯( linC β€˜π‘€)𝑉)))
47 simpr 484 . . . . . . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . . . 18 (𝑦( linC β€˜π‘€)𝑉) = (𝑦( linC β€˜π‘€)𝑉)
49 eqid 2726 . . . . . . . . . . . . . . . . . 18 (π‘₯( linC β€˜π‘€)𝑉) = (π‘₯( linC β€˜π‘€)𝑉)
5012, 48, 49, 2, 3, 25lincsum 47385 . . . . . . . . . . . . . . . . 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 2766 . . . . . . . . . . . . . . 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 5144 . . . . . . . . . . . . . . . . 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 7412 . . . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . 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 3606 . . . . . . . . . . . . . . 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 834 . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . 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 415 . . . . . . . . . . 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 407 . . . . . . . . . 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 407 . . . . . . 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 407 . . . . . 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 407 . . . . 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 47368 . . . . . 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 480 . . . . 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 710 . . . 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 412 . . 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 406 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 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064  π’« cpw 4597   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405   ∘f cof 7665   ↑m cmap 8822   finSupp cfsupp 9363  Basecbs 17153  +gcplusg 17206  Scalarcsca 17209  0gc0g 17394  Mndcmnd 18667  LModclmod 20706   linC clinc 47360   LinCo clinco 47361
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14296  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-gsum 17397  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-grp 18866  df-minusg 18867  df-cntz 19233  df-cmn 19702  df-abl 19703  df-mgp 20040  df-ur 20087  df-ring 20140  df-lmod 20708  df-linc 47362  df-lco 47363
This theorem is referenced by:  lincsumscmcl  47389
  Copyright terms: Public domain W3C validator