Theorem lco0 48458

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 5294	. . 3 ⊢ ∅ ∈ 𝒫 (Base‘𝑀)
2		eqid 2731	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2731	. . . 4 ⊢ (Scalar‘𝑀) = (Scalar‘𝑀)
4		eqid 2731	. . . 4 ⊢ (Base‘(Scalar‘𝑀)) = (Base‘(Scalar‘𝑀))
5	2, 3, 4	lcoop 48442	. . 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 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 6835	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
8		map0e 8806	. . . . . . 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 8392	. . . . . 6 ⊢ 1o = {∅}
11	9, 10	eqtrdi 2782	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
12	11	rexeqdv 3293	. . . 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 48446	. . . . . . . 8 ⊢ (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
14	13	adantr 480	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
15	14	eqeq2d 2742	. . . . . 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 5245	. . . . . 6 ⊢ ∅ ∈ V
18		breq1 5094	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19		fvex 6835	. . . . . . . . . . 11 ⊢ (0g‘(Scalar‘𝑀)) ∈ V
20		0fsupp 9274	. . . . . . . . . . 11 ⊢ ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
21	19, 20	ax-mp 5	. . . . . . . . . 10 ⊢ ∅ finSupp (0g‘(Scalar‘𝑀))
22		0fi 8964	. . . . . . . . . 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 7353	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
26	25	eqeq2d 2742	. . . . . . . 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 4629	. . . . . 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 532	. . . . 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 3401	. 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 2731	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
36	2, 35	mndidcl 18654	. . 3 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
37		rabsn 4674	. . 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 2770	1 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g‘𝑀)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  Vcvv 3436  ∅c0 4283  𝒫 cpw 4550  {csn 4576   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  1_oc1o 8378   ↑_m cmap 8750  Fincfn 8869   finSupp cfsupp 9245  Basecbs 17117  Scalarcsca 17161  0_gc0g 17340  Mndcmnd 18639   linC clinc 48435   LinCo clinco 48436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-supp 8091  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-map 8752  df-en 8870  df-fin 8873  df-fsupp 9246  df-seq 13906  df-0g 17342  df-gsum 17343  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-linc 48437  df-lco 48438

This theorem is referenced by:  lcoel0  48459