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 ∅) = {(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 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))
5	2, 3, 4	lcoop 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)∅))})
6	1, 5	mpan2 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)
9	7, 8	mp1i 13	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅) = 1o)
10		df1o2 7856	. . . . . 6 ⊢ 1o = {∅}
11	9, 10	syl6eq 2830	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑𝑚 ∅) = {∅})
12	11	rexeqdv 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)∅) = (0g‘𝑀))
14	13	adantr 474	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
15	14	eqeq2d 2788	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (∅( linC ‘𝑀)∅) ↔ 𝑣 = (0g‘𝑀)))
16	15	anbi2d 622	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g‘𝑀))))
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)))
21	19, 20	ax-mp 5	. . . . . . . . . 10 ⊢ ∅ finSupp (0g‘(Scalar‘𝑀))
22		0fin 8476	. . . . . . . . . 10 ⊢ ∅ ∈ Fin
23	21, 22	2th 256	. . . . . . . . 9 ⊢ (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
24	18, 23	syl6bb 279	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25		oveq1 6929	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
26	25	eqeq2d 2788	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑣 = (𝑤( linC ‘𝑀)∅) ↔ 𝑣 = (∅( linC ‘𝑀)∅)))
27	24, 26	anbi12d 624	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
28	27	rexsng 4445	. . . . . 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 528	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (𝑣 = (0g‘𝑀) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g‘𝑀))))
32	16, 29, 31	3bitr4d 303	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∃𝑤 ∈ {∅} (𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ 𝑣 = (0g‘𝑀)))
33	12, 32	bitrd 271	. . 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 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 = (0g‘𝑀)})
35		eqid 2778	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
36	2, 35	mndidcl 17694	. . 3 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
37		rabsn 4488	. . 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 2818	1 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})

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  1_oc1o 7836   ↑_𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  Basecbs 16255  Scalarcsca 16341  0_gc0g 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