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Theorem lco0 48273
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 5362 . . 3 ∅ ∈ 𝒫 (Base‘𝑀)
2 eqid 2735 . . . 4 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2735 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2735 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
52, 3, 4lcoop 48257 . . 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 691 . 2 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))})
7 fvex 6920 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
8 map0e 8921 . . . . . . 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 8512 . . . . . 6 1o = {∅}
119, 10eqtrdi 2791 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
1211rexeqdv 3325 . . . 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 48261 . . . . . . . 8 (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g𝑀))
1413adantr 480 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g𝑀))
1514eqeq2d 2746 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (∅( linC ‘𝑀)∅) ↔ 𝑣 = (0g𝑀)))
1615anbi2d 630 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g𝑀))))
17 0ex 5313 . . . . . 6 ∅ ∈ V
18 breq1 5151 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19 fvex 6920 . . . . . . . . . . 11 (0g‘(Scalar‘𝑀)) ∈ V
20 0fsupp 9428 . . . . . . . . . . 11 ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
2119, 20ax-mp 5 . . . . . . . . . 10 ∅ finSupp (0g‘(Scalar‘𝑀))
22 0fi 9081 . . . . . . . . . 10 ∅ ∈ Fin
2321, 222th 264 . . . . . . . . 9 (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
2418, 23bitrdi 287 . . . . . . . 8 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25 oveq1 7438 . . . . . . . . 9 (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
2625eqeq2d 2746 . . . . . . . 8 (𝑤 = ∅ → (𝑣 = (𝑤( linC ‘𝑀)∅) ↔ 𝑣 = (∅( linC ‘𝑀)∅)))
2724, 26anbi12d 632 . . . . . . 7 (𝑤 = ∅ → ((𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
2827rexsng 4681 . . . . . 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 532 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (0g𝑀) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g𝑀))))
3216, 29, 313bitr4d 311 . . . 4 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g𝑀)))
3312, 32bitrd 279 . . 3 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g𝑀)))
3433rabbidva 3440 . 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 2735 . . . 4 (0g𝑀) = (0g𝑀)
362, 35mndidcl 18775 . . 3 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
37 rabsn 4726 . . 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 2779 1 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  cfv 6563  (class class class)co 7431  1oc1o 8498  m cmap 8865  Fincfn 8984   finSupp cfsupp 9399  Basecbs 17245  Scalarcsca 17301  0gc0g 17486  Mndcmnd 18760   linC clinc 48250   LinCo clinco 48251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-map 8867  df-en 8985  df-fin 8988  df-fsupp 9400  df-seq 14040  df-0g 17488  df-gsum 17489  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-linc 48252  df-lco 48253
This theorem is referenced by:  lcoel0  48274
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