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Theorem lcoel0 42542
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 (Base‘𝑀)) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))

Proof of Theorem lcoel0
Dummy variables 𝑠 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6239 . . . 4 (0g𝑀) ∈ V
21snid 4241 . . 3 (0g𝑀) ∈ {(0g𝑀)}
3 oveq2 6698 . . . 4 (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4 lmodgrp 18918 . . . . . 6 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5 grpmnd 17476 . . . . . 6 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6 lco0 42541 . . . . . 6 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
74, 5, 63syl 18 . . . . 5 (𝑀 ∈ LMod → (𝑀 LinCo ∅) = {(0g𝑀)})
87adantr 480 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo ∅) = {(0g𝑀)})
93, 8sylan9eq 2705 . . 3 ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 LinCo 𝑉) = {(0g𝑀)})
102, 9syl5eleqr 2737 . 2 ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
11 eqid 2651 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
12 eqid 2651 . . . . . 6 (0g𝑀) = (0g𝑀)
1311, 12lmod0vcl 18940 . . . . 5 (𝑀 ∈ LMod → (0g𝑀) ∈ (Base‘𝑀))
1413adantr 480 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g𝑀) ∈ (Base‘𝑀))
1514adantl 481 . . 3 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (Base‘𝑀))
16 eqid 2651 . . . . . 6 (Scalar‘𝑀) = (Scalar‘𝑀)
17 eqid 2651 . . . . . 6 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
18 eqidd 2652 . . . . . . 7 (𝑣 = 𝑤 → (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)))
1918cbvmptv 4783 . . . . . 6 (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) = (𝑤𝑉 ↦ (0g‘(Scalar‘𝑀)))
20 eqid 2651 . . . . . 6 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2111, 16, 17, 12, 19, 20lcoc0 42536 . . . . 5 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)))
2221adantl 481 . . . 4 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)))
23 simpl 472 . . . . . . . 8 (((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))) → (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉))
24 breq1 4688 . . . . . . . . . 10 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25 oveq1 6697 . . . . . . . . . . . 12 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉))
2625eqeq2d 2661 . . . . . . . . . . 11 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ (0g𝑀) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉)))
27 eqcom 2658 . . . . . . . . . . 11 ((0g𝑀) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) ↔ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀))
2826, 27syl6bb 276 . . . . . . . . . 10 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)))
2924, 28anbi12d 747 . . . . . . . . 9 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀))))
3029adantl 481 . . . . . . . 8 ((((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))) ∧ 𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))) → ((𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉)) ↔ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀))))
3123, 30rspcedv 3344 . . . . . . 7 (((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)))) → (((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉))))
3231ex 449 . . . . . 6 ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) → ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉)))))
3332com23 86 . . . . 5 ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) → (((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)) → ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉)))))
34333impib 1281 . . . 4 (((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀)) ∧ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)) → ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉))))
3522, 34mpcom 38 . . 3 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉)))
3611, 16, 20lcoval 42526 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → ((0g𝑀) ∈ (𝑀 LinCo 𝑉) ↔ ((0g𝑀) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉)))))
3736adantl 481 . . 3 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → ((0g𝑀) ∈ (𝑀 LinCo 𝑉) ↔ ((0g𝑀) ∈ (Base‘𝑀) ∧ ∃𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉)(𝑠 finSupp (0g‘(Scalar‘𝑀)) ∧ (0g𝑀) = (𝑠( linC ‘𝑀)𝑉)))))
3815, 35, 37mpbir2and 977 . 2 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
3910, 38pm2.61ian 848 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  c0 3948  𝒫 cpw 4191  {csn 4210   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  𝑚 cmap 7899   finSupp cfsupp 8316  Basecbs 15904  Scalarcsca 15991  0gc0g 16147  Mndcmnd 17341  Grpcgrp 17469  LModclmod 18911   linC clinc 42518   LinCo clinco 42519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-map 7901  df-en 7998  df-fin 8001  df-fsupp 8317  df-seq 12842  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-ring 18595  df-lmod 18913  df-linc 42520  df-lco 42521
This theorem is referenced by:  lincolss  42548
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