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Theorem lcoel0 46499
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 6855 . . . 4 (0g𝑀) ∈ V
21snid 4622 . . 3 (0g𝑀) ∈ {(0g𝑀)}
3 oveq2 7365 . . . 4 (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4 lmodgrp 20329 . . . . . 6 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5 grpmnd 18755 . . . . . 6 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6 lco0 46498 . . . . . 6 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
74, 5, 63syl 18 . . . . 5 (𝑀 ∈ LMod → (𝑀 LinCo ∅) = {(0g𝑀)})
87adantr 481 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo ∅) = {(0g𝑀)})
93, 8sylan9eq 2796 . . 3 ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 LinCo 𝑉) = {(0g𝑀)})
102, 9eleqtrrid 2845 . 2 ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
11 eqid 2736 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
12 eqid 2736 . . . . . 6 (0g𝑀) = (0g𝑀)
1311, 12lmod0vcl 20351 . . . . 5 (𝑀 ∈ LMod → (0g𝑀) ∈ (Base‘𝑀))
1413adantr 481 . . . 4 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g𝑀) ∈ (Base‘𝑀))
1514adantl 482 . . 3 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (Base‘𝑀))
16 eqid 2736 . . . . . 6 (Scalar‘𝑀) = (Scalar‘𝑀)
17 eqid 2736 . . . . . 6 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
18 eqidd 2737 . . . . . . 7 (𝑣 = 𝑤 → (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)))
1918cbvmptv 5218 . . . . . 6 (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) = (𝑤𝑉 ↦ (0g‘(Scalar‘𝑀)))
20 eqid 2736 . . . . . 6 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2111, 16, 17, 12, 19, 20lcoc0 46493 . . . . 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 482 . . . 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 483 . . . . . . . 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 5108 . . . . . . . . . 10 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25 oveq1 7364 . . . . . . . . . . . 12 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉))
2625eqeq2d 2747 . . . . . . . . . . 11 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ (0g𝑀) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉)))
27 eqcom 2743 . . . . . . . . . . 11 ((0g𝑀) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) ↔ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀))
2826, 27bitrdi 286 . . . . . . . . . 10 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)))
2924, 28anbi12d 631 . . . . . . . . 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 482 . . . . . . . 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 3574 . . . . . . 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 413 . . . . . 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 1116 . . . 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 46483 . . . 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 482 . . 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 711 . 2 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
3910, 38pm2.61ian 810 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  c0 4282  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cmpt 5188  cfv 6496  (class class class)co 7357  m cmap 8765   finSupp cfsupp 9305  Basecbs 17083  Scalarcsca 17136  0gc0g 17321  Mndcmnd 18556  Grpcgrp 18748  LModclmod 20322   linC clinc 46475   LinCo clinco 46476
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-map 8767  df-en 8884  df-fin 8887  df-fsupp 9306  df-seq 13907  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-ring 19966  df-lmod 20324  df-linc 46477  df-lco 46478
This theorem is referenced by:  lincolss  46505
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