Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcoel0 Structured version   Visualization version   GIF version

Theorem lcoel0 47062
Description: The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
Assertion
Ref Expression
lcoel0 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (0gβ€˜π‘€) ∈ (𝑀 LinCo 𝑉))

Proof of Theorem lcoel0
Dummy variables 𝑠 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6901 . . . 4 (0gβ€˜π‘€) ∈ V
21snid 4663 . . 3 (0gβ€˜π‘€) ∈ {(0gβ€˜π‘€)}
3 oveq2 7413 . . . 4 (𝑉 = βˆ… β†’ (𝑀 LinCo 𝑉) = (𝑀 LinCo βˆ…))
4 lmodgrp 20470 . . . . . 6 (𝑀 ∈ LMod β†’ 𝑀 ∈ Grp)
5 grpmnd 18822 . . . . . 6 (𝑀 ∈ Grp β†’ 𝑀 ∈ Mnd)
6 lco0 47061 . . . . . 6 (𝑀 ∈ Mnd β†’ (𝑀 LinCo βˆ…) = {(0gβ€˜π‘€)})
74, 5, 63syl 18 . . . . 5 (𝑀 ∈ LMod β†’ (𝑀 LinCo βˆ…) = {(0gβ€˜π‘€)})
87adantr 481 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (𝑀 LinCo βˆ…) = {(0gβ€˜π‘€)})
93, 8sylan9eq 2792 . . 3 ((𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (𝑀 LinCo 𝑉) = {(0gβ€˜π‘€)})
102, 9eleqtrrid 2840 . 2 ((𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (0gβ€˜π‘€) ∈ (𝑀 LinCo 𝑉))
11 eqid 2732 . . . . . 6 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
12 eqid 2732 . . . . . 6 (0gβ€˜π‘€) = (0gβ€˜π‘€)
1311, 12lmod0vcl 20493 . . . . 5 (𝑀 ∈ LMod β†’ (0gβ€˜π‘€) ∈ (Baseβ€˜π‘€))
1413adantr 481 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (0gβ€˜π‘€) ∈ (Baseβ€˜π‘€))
1514adantl 482 . . 3 ((Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (0gβ€˜π‘€) ∈ (Baseβ€˜π‘€))
16 eqid 2732 . . . . . 6 (Scalarβ€˜π‘€) = (Scalarβ€˜π‘€)
17 eqid 2732 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜(Scalarβ€˜π‘€))
18 eqidd 2733 . . . . . . 7 (𝑣 = 𝑀 β†’ (0gβ€˜(Scalarβ€˜π‘€)) = (0gβ€˜(Scalarβ€˜π‘€)))
1918cbvmptv 5260 . . . . . 6 (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) = (𝑀 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))
20 eqid 2732 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘€)) = (Baseβ€˜(Scalarβ€˜π‘€))
2111, 16, 17, 12, 19, 20lcoc0 47056 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€)))
2221adantl 482 . . . 4 ((Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€)))
23 simpl 483 . . . . . . . 8 (((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))) β†’ (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉))
24 breq1 5150 . . . . . . . . . 10 (𝑠 = (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) β†’ (𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ↔ (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€))))
25 oveq1 7412 . . . . . . . . . . . 12 (𝑠 = (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) β†’ (𝑠( linC β€˜π‘€)𝑉) = ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉))
2625eqeq2d 2743 . . . . . . . . . . 11 (𝑠 = (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) β†’ ((0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉) ↔ (0gβ€˜π‘€) = ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉)))
27 eqcom 2739 . . . . . . . . . . 11 ((0gβ€˜π‘€) = ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) ↔ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€))
2826, 27bitrdi 286 . . . . . . . . . 10 (𝑠 = (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) β†’ ((0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉) ↔ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€)))
2924, 28anbi12d 631 . . . . . . . . 9 (𝑠 = (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) β†’ ((𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉)) ↔ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€))))
3029adantl 482 . . . . . . . 8 ((((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))) ∧ 𝑠 = (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))) β†’ ((𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉)) ↔ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€))))
3123, 30rspcedv 3605 . . . . . . 7 (((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)))) β†’ (((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉))))
3231ex 413 . . . . . 6 ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) β†’ ((Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€)) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉)))))
3332com23 86 . . . . 5 ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) β†’ (((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€)) β†’ ((Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉)))))
34333impib 1116 . . . 4 (((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉) ∧ (𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€))) finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ ((𝑣 ∈ 𝑉 ↦ (0gβ€˜(Scalarβ€˜π‘€)))( linC β€˜π‘€)𝑉) = (0gβ€˜π‘€)) β†’ ((Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉))))
3522, 34mpcom 38 . . 3 ((Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉)))
3611, 16, 20lcoval 47046 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ ((0gβ€˜π‘€) ∈ (𝑀 LinCo 𝑉) ↔ ((0gβ€˜π‘€) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉)))))
3736adantl 482 . . 3 ((Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ ((0gβ€˜π‘€) ∈ (𝑀 LinCo 𝑉) ↔ ((0gβ€˜π‘€) ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘  ∈ ((Baseβ€˜(Scalarβ€˜π‘€)) ↑m 𝑉)(𝑠 finSupp (0gβ€˜(Scalarβ€˜π‘€)) ∧ (0gβ€˜π‘€) = (𝑠( linC β€˜π‘€)𝑉)))))
3815, 35, 37mpbir2and 711 . 2 ((Β¬ 𝑉 = βˆ… ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€))) β†’ (0gβ€˜π‘€) ∈ (𝑀 LinCo 𝑉))
3910, 38pm2.61ian 810 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) β†’ (0gβ€˜π‘€) ∈ (𝑀 LinCo 𝑉))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  βˆ…c0 4321  π’« cpw 4601  {csn 4627   class class class wbr 5147   ↦ cmpt 5230  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816   finSupp cfsupp 9357  Basecbs 17140  Scalarcsca 17196  0gc0g 17381  Mndcmnd 18621  Grpcgrp 18815  LModclmod 20463   linC clinc 47038   LinCo clinco 47039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-map 8818  df-en 8936  df-fin 8939  df-fsupp 9358  df-seq 13963  df-0g 17383  df-gsum 17384  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-ring 20051  df-lmod 20465  df-linc 47040  df-lco 47041
This theorem is referenced by:  lincolss  47068
  Copyright terms: Public domain W3C validator