Theorem lco0 45768

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 5278	. . 3 ⊢ ∅ ∈ 𝒫 (Base‘𝑀)
2		eqid 2738	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2738	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
4		eqid 2738	. . . 4 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5	2, 3, 4	lcoop 45752	. . 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 688	. 2 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))})
7		fvex 6787	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
8		map0e 8670	. . . . . . 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 8304	. . . . . 6 ⊢ 1o = {∅}
11	9, 10	eqtrdi 2794	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
12	11	rexeqdv 3349	. . . 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 45756	. . . . . . . 8 ⊢ (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
14	13	adantr 481	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
15	14	eqeq2d 2749	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (∅( linC ‘𝑀)∅) ↔ 𝑣 = (0g‘𝑀)))
16	15	anbi2d 629	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g‘𝑀))))
17		0ex 5231	. . . . . 6 ⊢ ∅ ∈ V
18		breq1 5077	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19		fvex 6787	. . . . . . . . . . 11 ⊢ (0g‘(Scalar‘𝑀)) ∈ V
20		0fsupp 9150	. . . . . . . . . . 11 ⊢ ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
21	19, 20	ax-mp 5	. . . . . . . . . 10 ⊢ ∅ finSupp (0g‘(Scalar‘𝑀))
22		0fin 8954	. . . . . . . . . 10 ⊢ ∅ ∈ Fin
23	21, 22	2th 263	. . . . . . . . 9 ⊢ (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
24	18, 23	bitrdi 287	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25		oveq1 7282	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
26	25	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑣 = (𝑤( linC ‘𝑀)∅) ↔ 𝑣 = (∅( linC ‘𝑀)∅)))
27	24, 26	anbi12d 631	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
28	27	rexsng 4610	. . . . . 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 533	. . . . 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 278	. . 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 3413	. 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 2738	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
36	2, 35	mndidcl 18400	. . 3 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
37		rabsn 4657	. . 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 2782	1 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {crab 3068  Vcvv 3432  ∅c0 4256  𝒫 cpw 4533  {csn 4561   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290   ↑_m cmap 8615  Fincfn 8733   finSupp cfsupp 9128  Basecbs 16912  Scalarcsca 16965  0_gc0g 17150  Mndcmnd 18385   linC clinc 45745   LinCo clinco 45746

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-map 8617  df-en 8734  df-fin 8737  df-fsupp 9129  df-seq 13722  df-0g 17152  df-gsum 17153  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-linc 45747  df-lco 45748

This theorem is referenced by:  lcoel0  45769