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 48345
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 6919 . . . 4 (0g𝑀) ∈ V
21snid 4662 . . 3 (0g𝑀) ∈ {(0g𝑀)}
3 oveq2 7439 . . . 4 (𝑉 = ∅ → (𝑀 LinCo 𝑉) = (𝑀 LinCo ∅))
4 lmodgrp 20865 . . . . . 6 (𝑀 ∈ LMod → 𝑀 ∈ Grp)
5 grpmnd 18958 . . . . . 6 (𝑀 ∈ Grp → 𝑀 ∈ Mnd)
6 lco0 48344 . . . . . 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 2797 . . 3 ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (𝑀 LinCo 𝑉) = {(0g𝑀)})
102, 9eleqtrrid 2848 . 2 ((𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
11 eqid 2737 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
12 eqid 2737 . . . . . 6 (0g𝑀) = (0g𝑀)
1311, 12lmod0vcl 20889 . . . . 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 2737 . . . . . 6 (Scalar‘𝑀) = (Scalar‘𝑀)
17 eqid 2737 . . . . . 6 (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀))
18 eqidd 2738 . . . . . . 7 (𝑣 = 𝑤 → (0g‘(Scalar‘𝑀)) = (0g‘(Scalar‘𝑀)))
1918cbvmptv 5255 . . . . . 6 (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) = (𝑤𝑉 ↦ (0g‘(Scalar‘𝑀)))
20 eqid 2737 . . . . . 6 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
2111, 16, 17, 12, 19, 20lcoc0 48339 . . . . 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 482 . . . . . . . 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 5146 . . . . . . . . . 10 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠 finSupp (0g‘(Scalar‘𝑀)) ↔ (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) finSupp (0g‘(Scalar‘𝑀))))
25 oveq1 7438 . . . . . . . . . . . 12 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → (𝑠( linC ‘𝑀)𝑉) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉))
2625eqeq2d 2748 . . . . . . . . . . 11 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ (0g𝑀) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉)))
27 eqcom 2744 . . . . . . . . . . 11 ((0g𝑀) = ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) ↔ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀))
2826, 27bitrdi 287 . . . . . . . . . 10 (𝑠 = (𝑣𝑉 ↦ (0g‘(Scalar‘𝑀))) → ((0g𝑀) = (𝑠( linC ‘𝑀)𝑉) ↔ ((𝑣𝑉 ↦ (0g‘(Scalar‘𝑀)))( linC ‘𝑀)𝑉) = (0g𝑀)))
2924, 28anbi12d 632 . . . . . . . . 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 3615 . . . . . . 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 412 . . . . . 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 1117 . . . 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 48329 . . . 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 713 . 2 ((¬ 𝑉 = ∅ ∧ (𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀))) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
3910, 38pm2.61ian 812 1 ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  m cmap 8866   finSupp cfsupp 9401  Basecbs 17247  Scalarcsca 17300  0gc0g 17484  Mndcmnd 18747  Grpcgrp 18951  LModclmod 20858   linC clinc 48321   LinCo clinco 48322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-map 8868  df-en 8986  df-fin 8989  df-fsupp 9402  df-seq 14043  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ring 20232  df-lmod 20860  df-linc 48323  df-lco 48324
This theorem is referenced by:  lincolss  48351
  Copyright terms: Public domain W3C validator