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Theorem lco0 43231
 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 5068 . . 3 ∅ ∈ 𝒫 (Base‘𝑀)
2 eqid 2778 . . . 4 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2778 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2778 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
52, 3, 4lcoop 43215 . . 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 681 . 2 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))})
7 fvex 6459 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
8 map0e 8179 . . . . . . 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 7856 . . . . . 6 1o = {∅}
119, 10syl6eq 2830 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅) = {∅})
1211rexeqdv 3341 . . . 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 43219 . . . . . . . 8 (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g𝑀))
1413adantr 474 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g𝑀))
1514eqeq2d 2788 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (∅( linC ‘𝑀)∅) ↔ 𝑣 = (0g𝑀)))
1615anbi2d 622 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g𝑀))))
17 0ex 5026 . . . . . 6 ∅ ∈ V
18 breq1 4889 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19 fvex 6459 . . . . . . . . . . 11 (0g‘(Scalar‘𝑀)) ∈ V
20 0fsupp 8585 . . . . . . . . . . 11 ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
2119, 20ax-mp 5 . . . . . . . . . 10 ∅ finSupp (0g‘(Scalar‘𝑀))
22 0fin 8476 . . . . . . . . . 10 ∅ ∈ Fin
2321, 222th 256 . . . . . . . . 9 (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
2418, 23syl6bb 279 . . . . . . . 8 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25 oveq1 6929 . . . . . . . . 9 (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
2625eqeq2d 2788 . . . . . . . 8 (𝑤 = ∅ → (𝑣 = (𝑤( linC ‘𝑀)∅) ↔ 𝑣 = (∅( linC ‘𝑀)∅)))
2724, 26anbi12d 624 . . . . . . 7 (𝑤 = ∅ → ((𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
2827rexsng 4445 . . . . . 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 528 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (0g𝑀) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g𝑀))))
3216, 29, 313bitr4d 303 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g𝑀)))
3312, 32bitrd 271 . . 3 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g𝑀)))
3433rabbidva 3385 . 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 2778 . . . 4 (0g𝑀) = (0g𝑀)
362, 35mndidcl 17694 . . 3 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
37 rabsn 4488 . . 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 2818 1 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∃wrex 3091  {crab 3094  Vcvv 3398  ∅c0 4141  𝒫 cpw 4379  {csn 4398   class class class wbr 4886  ‘cfv 6135  (class class class)co 6922  1oc1o 7836   ↑𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  Basecbs 16255  Scalarcsca 16341  0gc0g 16486  Mndcmnd 17680   linC clinc 43208   LinCo clinco 43209 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-map 8142  df-en 8242  df-fin 8245  df-fsupp 8564  df-seq 13120  df-0g 16488  df-gsum 16489  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-linc 43210  df-lco 43211 This theorem is referenced by:  lcoel0  43232
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