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Theorem lco0 46498
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 5311 . . 3 ∅ ∈ 𝒫 (Base‘𝑀)
2 eqid 2736 . . . 4 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2736 . . . 4 (Scalar‘𝑀) = (Scalar‘𝑀)
4 eqid 2736 . . . 4 (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
52, 3, 4lcoop 46482 . . 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 6855 . . . . . . 7 (Base‘(Scalar‘𝑀)) ∈ V
8 map0e 8820 . . . . . . 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 8419 . . . . . 6 1o = {∅}
119, 10eqtrdi 2792 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
1211rexeqdv 3314 . . . 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 46486 . . . . . . . 8 (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g𝑀))
1413adantr 481 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g𝑀))
1514eqeq2d 2747 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (∅( linC ‘𝑀)∅) ↔ 𝑣 = (0g𝑀)))
1615anbi2d 629 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g𝑀))))
17 0ex 5264 . . . . . 6 ∅ ∈ V
18 breq1 5108 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19 fvex 6855 . . . . . . . . . . 11 (0g‘(Scalar‘𝑀)) ∈ V
20 0fsupp 9327 . . . . . . . . . . 11 ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
2119, 20ax-mp 5 . . . . . . . . . 10 ∅ finSupp (0g‘(Scalar‘𝑀))
22 0fin 9115 . . . . . . . . . 10 ∅ ∈ Fin
2321, 222th 263 . . . . . . . . 9 (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
2418, 23bitrdi 286 . . . . . . . 8 (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25 oveq1 7364 . . . . . . . . 9 (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
2625eqeq2d 2747 . . . . . . . 8 (𝑤 = ∅ → (𝑣 = (𝑤( linC ‘𝑀)∅) ↔ 𝑣 = (∅( linC ‘𝑀)∅)))
2724, 26anbi12d 631 . . . . . . 7 (𝑤 = ∅ → ((𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
2827rexsng 4635 . . . . . 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 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 3414 . 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 2736 . . . 4 (0g𝑀) = (0g𝑀)
362, 35mndidcl 18571 . . 3 (𝑀 ∈ Mnd → (0g𝑀) ∈ (Base‘𝑀))
37 rabsn 4682 . . 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 2780 1 (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  {crab 3407  Vcvv 3445  c0 4282  𝒫 cpw 4560  {csn 4586   class class class wbr 5105  cfv 6496  (class class class)co 7357  1oc1o 8405  m cmap 8765  Fincfn 8883   finSupp cfsupp 9305  Basecbs 17083  Scalarcsca 17136  0gc0g 17321  Mndcmnd 18556   linC clinc 46475   LinCo clinco 46476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-map 8767  df-en 8884  df-fin 8887  df-fsupp 9306  df-seq 13907  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-linc 46477  df-lco 46478
This theorem is referenced by:  lcoel0  46499
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