Theorem lco0 47108

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 ∅) = {(0g‘𝑀)})

Proof of Theorem lco0

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 5355	. . 3 ⊢ ∅ ∈ 𝒫 (Base‘𝑀)
2		eqid 2733	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2733	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
4		eqid 2733	. . . 4 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5	2, 3, 4	lcoop 47092	. . 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)∅))})
6	1, 5	mpan2 690	. 2 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))})
7		fvex 6905	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
8		map0e 8876	. . . . . . 7 ⊢ ((Base‘(Scalar‘𝑀)) ∈ V → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
9	7, 8	mp1i 13	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = 1o)
10		df1o2 8473	. . . . . 6 ⊢ 1o = {∅}
11	9, 10	eqtrdi 2789	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
12	11	rexeqdv 3327	. . . 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 47096	. . . . . . . 8 ⊢ (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
14	13	adantr 482	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
15	14	eqeq2d 2744	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (∅( linC ‘𝑀)∅) ↔ 𝑣 = (0g‘𝑀)))
16	15	anbi2d 630	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g‘𝑀))))
17		0ex 5308	. . . . . 6 ⊢ ∅ ∈ V
18		breq1 5152	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19		fvex 6905	. . . . . . . . . . 11 ⊢ (0g‘(Scalar‘𝑀)) ∈ V
20		0fsupp 9385	. . . . . . . . . . 11 ⊢ ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
21	19, 20	ax-mp 5	. . . . . . . . . 10 ⊢ ∅ finSupp (0g‘(Scalar‘𝑀))
22		0fin 9171	. . . . . . . . . 10 ⊢ ∅ ∈ Fin
23	21, 22	2th 264	. . . . . . . . 9 ⊢ (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
24	18, 23	bitrdi 287	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25		oveq1 7416	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
26	25	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑣 = (𝑤( linC ‘𝑀)∅) ↔ 𝑣 = (∅( linC ‘𝑀)∅)))
27	24, 26	anbi12d 632	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
28	27	rexsng 4679	. . . . . 6 ⊢ (∅ ∈ V → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
29	17, 28	mp1i 13	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
30	22	a1i 11	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ∅ ∈ Fin)
31	30	biantrurd 534	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (0g‘𝑀) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g‘𝑀))))
32	16, 29, 31	3bitr4d 311	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g‘𝑀)))
33	12, 32	bitrd 279	. . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g‘𝑀)))
34	33	rabbidva 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 = (0g‘𝑀)})
35		eqid 2733	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
36	2, 35	mndidcl 18640	. . 3 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
37		rabsn 4726	. . 3 ⊢ ((0g‘𝑀) ∈ (Base‘𝑀) → {𝑣 ∈ (Base‘𝑀) ∣ 𝑣 = (0g‘𝑀)} = {(0g‘𝑀)})
38	36, 37	syl 17	. 2 ⊢ (𝑀 ∈ Mnd → {𝑣 ∈ (Base‘𝑀) ∣ 𝑣 = (0g‘𝑀)} = {(0g‘𝑀)})
39	6, 34, 38	3eqtrd 2777	1 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  {crab 3433  Vcvv 3475  ∅c0 4323  𝒫 cpw 4603  {csn 4629   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  1_oc1o 8459   ↑_m cmap 8820  Fincfn 8939   finSupp cfsupp 9361  Basecbs 17144  Scalarcsca 17200  0_gc0g 17385  Mndcmnd 18625   linC clinc 47085   LinCo clinco 47086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-map 8822  df-en 8940  df-fin 8943  df-fsupp 9362  df-seq 13967  df-0g 17387  df-gsum 17388  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-linc 47087  df-lco 47088

This theorem is referenced by:  lcoel0  47109