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Theorem lco0 44410
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 5247 . . 3 ∅ ∈ 𝒫 (Base‘𝑀)
2 eqid 2818 . . . 4 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2818 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2818 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
52, 3, 4lcoop 44394 . . 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 687 . 2 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))})
7 fvex 6676 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
8 map0e 8435 . . . . . . 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 8105 . . . . . 6 1o = {∅}
119, 10syl6eq 2869 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
1211rexeqdv 3414 . . . 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 44398 . . . . . . . 8 (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g𝑀))
1413adantr 481 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g𝑀))
1514eqeq2d 2829 . . . . . 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 5202 . . . . . 6 ∅ ∈ V
18 breq1 5060 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19 fvex 6676 . . . . . . . . . . 11 (0g‘(Scalar‘𝑀)) ∈ V
20 0fsupp 8843 . . . . . . . . . . 11 ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
2119, 20ax-mp 5 . . . . . . . . . 10 ∅ finSupp (0g‘(Scalar‘𝑀))
22 0fin 8734 . . . . . . . . . 10 ∅ ∈ Fin
2321, 222th 265 . . . . . . . . 9 (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
2418, 23syl6bb 288 . . . . . . . 8 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25 oveq1 7152 . . . . . . . . 9 (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
2625eqeq2d 2829 . . . . . . . 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 4606 . . . . . 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 533 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (0g𝑀) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g𝑀))))
3216, 29, 313bitr4d 312 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g𝑀)))
3312, 32bitrd 280 . . 3 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g𝑀)))
3433rabbidva 3476 . 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 2818 . . . 4 (0g𝑀) = (0g𝑀)
362, 35mndidcl 17914 . . 3 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
37 rabsn 4649 . . 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 2857 1 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  {crab 3139  Vcvv 3492  c0 4288  𝒫 cpw 4535  {csn 4557   class class class wbr 5057  cfv 6348  (class class class)co 7145  1oc1o 8084  m cmap 8395  Fincfn 8497   finSupp cfsupp 8821  Basecbs 16471  Scalarcsca 16556  0gc0g 16701  Mndcmnd 17899   linC clinc 44387   LinCo clinco 44388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-map 8397  df-en 8498  df-fin 8501  df-fsupp 8822  df-seq 13358  df-0g 16703  df-gsum 16704  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-linc 44389  df-lco 44390
This theorem is referenced by:  lcoel0  44411
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