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Theorem lco0 47681
Description: The set of empty linear combinations over a monoid is the singleton with the identity element of the monoid. (Contributed by AV, 12-Apr-2019.)
Assertion
Ref Expression
lco0 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})

Proof of Theorem lco0
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elpw 5356 . . 3 ∅ ∈ 𝒫 (Base‘𝑀)
2 eqid 2725 . . . 4 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2725 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2725 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
52, 3, 4lcoop 47665 . . 3 ((𝑀 ∈ Mnd ∧ ∅ ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo ∅) = {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))})
61, 5mpan2 689 . 2 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))})
7 fvex 6909 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
8 map0e 8901 . . . . . . 7 ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
97, 8mp1i 13 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
10 df1o2 8494 . . . . . 6 1o = {∅}
119, 10eqtrdi 2781 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
1211rexeqdv 3315 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ ∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))))
13 lincval0 47669 . . . . . . . 8 (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g𝑀))
1413adantr 479 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g𝑀))
1514eqeq2d 2736 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (∅( linC ‘𝑀)∅) ↔ 𝑣 = (0g𝑀)))
1615anbi2d 628 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g𝑀))))
17 0ex 5308 . . . . . 6 ∅ ∈ V
18 breq1 5152 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19 fvex 6909 . . . . . . . . . . 11 (0g‘(Scalar‘𝑀)) ∈ V
20 0fsupp 9415 . . . . . . . . . . 11 ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
2119, 20ax-mp 5 . . . . . . . . . 10 ∅ finSupp (0g‘(Scalar‘𝑀))
22 0fin 9196 . . . . . . . . . 10 ∅ ∈ Fin
2321, 222th 263 . . . . . . . . 9 (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
2418, 23bitrdi 286 . . . . . . . 8 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25 oveq1 7426 . . . . . . . . 9 (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
2625eqeq2d 2736 . . . . . . . 8 (𝑤 = ∅ → (𝑣 = (𝑤( linC ‘𝑀)∅) ↔ 𝑣 = (∅( linC ‘𝑀)∅)))
2724, 26anbi12d 630 . . . . . . 7 (𝑤 = ∅ → ((𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
2827rexsng 4680 . . . . . 6 (∅ ∈ V → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
2917, 28mp1i 13 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
3022a1i 11 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ∅ ∈ Fin)
3130biantrurd 531 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (0g𝑀) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g𝑀))))
3216, 29, 313bitr4d 310 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g𝑀)))
3312, 32bitrd 278 . . 3 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g𝑀)))
3433rabbidva 3425 . 2 (𝑀 ∈ Mnd → {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))} = {𝑣 ∈ (Base‘𝑀) ∣ 𝑣 = (0g𝑀)})
35 eqid 2725 . . . 4 (0g𝑀) = (0g𝑀)
362, 35mndidcl 18712 . . 3 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
37 rabsn 4727 . . 3 ((0g𝑀) ∈ (Base‘𝑀) → {𝑣 ∈ (Base‘𝑀) ∣ 𝑣 = (0g𝑀)} = {(0g𝑀)})
3836, 37syl 17 . 2 (𝑀 ∈ Mnd → {𝑣 ∈ (Base‘𝑀) ∣ 𝑣 = (0g𝑀)} = {(0g𝑀)})
396, 34, 383eqtrd 2769 1 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3059  {crab 3418  Vcvv 3461  c0 4322  𝒫 cpw 4604  {csn 4630   class class class wbr 5149  cfv 6549  (class class class)co 7419  1oc1o 8480  m cmap 8845  Fincfn 8964   finSupp cfsupp 9387  Basecbs 17183  Scalarcsca 17239  0gc0g 17424  Mndcmnd 18697   linC clinc 47658   LinCo clinco 47659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-map 8847  df-en 8965  df-fin 8968  df-fsupp 9388  df-seq 14003  df-0g 17426  df-gsum 17427  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-linc 47660  df-lco 47661
This theorem is referenced by:  lcoel0  47682
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