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Theorem lcoel0 42719
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 6354 . . . 4 (0g𝑀) ∈ V
21snid 4345 . . 3 (0g𝑀) ∈ {(0g𝑀)}
3 oveq2 6813 . . . 4 (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4 lmodgrp 19064 . . . . . 6 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5 grpmnd 17622 . . . . . 6 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6 lco0 42718 . . . . . 6 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
74, 5, 63syl 18 . . . . 5 (𝑀 ∈ LMod → (𝑀 LinCo ∅) = {(0g𝑀)})
87adantr 472 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo ∅) = {(0g𝑀)})
93, 8sylan9eq 2806 . . 3 ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 LinCo 𝑉) = {(0g𝑀)})
102, 9syl5eleqr 2838 . 2 ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
11 eqid 2752 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
12 eqid 2752 . . . . . 6 (0g𝑀) = (0g𝑀)
1311, 12lmod0vcl 19086 . . . . 5 (𝑀 ∈ LMod → (0g𝑀) ∈ (Base‘𝑀))
1413adantr 472 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g𝑀) ∈ (Base‘𝑀))
1514adantl 473 . . 3 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (Base‘𝑀))
16 eqid 2752 . . . . . 6 (Scalar‘𝑀) = (Scalar‘𝑀)
17 eqid 2752 . . . . . 6 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
18 eqidd 2753 . . . . . . 7 (𝑣 = 𝑤 → (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)))
1918cbvmptv 4894 . . . . . 6 (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) = (𝑤𝑉 ↦ (0g‘(Scalar‘𝑀)))
20 eqid 2752 . . . . . 6 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2111, 16, 17, 12, 19, 20lcoc0 42713 . . . . 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 473 . . . 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 474 . . . . . . . 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 4799 . . . . . . . . . 10 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25 oveq1 6812 . . . . . . . . . . . 12 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉))
2625eqeq2d 2762 . . . . . . . . . . 11 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ (0g𝑀) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉)))
27 eqcom 2759 . . . . . . . . . . 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 749 . . . . . . . . 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 473 . . . . . . . 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 3445 . . . . . . 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 1108 . . . 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 42703 . . . 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 473 . . 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 995 . 2 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
3910, 38pm2.61ian 866 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 1072   = wceq 1624  wcel 2131  wrex 3043  c0 4050  𝒫 cpw 4294  {csn 4313   class class class wbr 4796  cmpt 4873  cfv 6041  (class class class)co 6805  𝑚 cmap 8015   finSupp cfsupp 8432  Basecbs 16051  Scalarcsca 16138  0gc0g 16294  Mndcmnd 17487  Grpcgrp 17615  LModclmod 19057   linC clinc 42695   LinCo clinco 42696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-supp 7456  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-map 8017  df-en 8114  df-fin 8117  df-fsupp 8433  df-seq 12988  df-0g 16296  df-gsum 16297  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-ring 18741  df-lmod 19059  df-linc 42697  df-lco 42698
This theorem is referenced by:  lincolss  42725
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