Theorem lco0 45656

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 5273	. . 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 45640	. . 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 687	. 2 ⊢ (𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {𝑣 ∈ (Base‘𝑀) ∣ ∃𝑤 ∈ ((Base‘(Scalar‘𝑀)) ↑m ∅)(𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅))})
7		fvex 6769	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑀)) ∈ V
8		map0e 8628	. . . . . . 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 8279	. . . . . 6 ⊢ 1o = {∅}
11	9, 10	eqtrdi 2795	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((Base‘(Scalar‘𝑀)) ↑m ∅) = {∅})
12	11	rexeqdv 3340	. . . 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 45644	. . . . . . . 8 ⊢ (𝑀 ∈ Mnd → (∅( linC ‘𝑀)∅) = (0g‘𝑀))
14	13	adantr 480	. . . . . . 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 628	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝑣 ∈ (Base‘𝑀)) → ((∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (0g‘𝑀))))
17		0ex 5226	. . . . . 6 ⊢ ∅ ∈ V
18		breq1 5073	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ finSupp (0g‘(Scalar‘𝑀))))
19		fvex 6769	. . . . . . . . . . 11 ⊢ (0g‘(Scalar‘𝑀)) ∈ V
20		0fsupp 9080	. . . . . . . . . . 11 ⊢ ((0g‘(Scalar‘𝑀)) ∈ V → ∅ finSupp (0g‘(Scalar‘𝑀)))
21	19, 20	ax-mp 5	. . . . . . . . . 10 ⊢ ∅ finSupp (0g‘(Scalar‘𝑀))
22		0fin 8916	. . . . . . . . . 10 ⊢ ∅ ∈ Fin
23	21, 22	2th 263	. . . . . . . . 9 ⊢ (∅ finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin)
24	18, 23	bitrdi 286	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑤 finSupp (0g‘(Scalar‘𝑀)) ↔ ∅ ∈ Fin))
25		oveq1 7262	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑤( linC ‘𝑀)∅) = (∅( linC ‘𝑀)∅))
26	25	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑣 = (𝑤( linC ‘𝑀)∅) ↔ 𝑣 = (∅( linC ‘𝑀)∅)))
27	24, 26	anbi12d 630	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑤 finSupp (0g‘(Scalar‘𝑀)) ∧ 𝑣 = (𝑤( linC ‘𝑀)∅)) ↔ (∅ ∈ Fin ∧ 𝑣 = (∅( linC ‘𝑀)∅))))
28	27	rexsng 4607	. . . . . 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 310	. . . 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 3402	. 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 18315	. . 3 ⊢ (𝑀 ∈ Mnd → (0g‘𝑀) ∈ (Base‘𝑀))
37		rabsn 4654	. . 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 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  {crab 3067  Vcvv 3422  ∅c0 4253  𝒫 cpw 4530  {csn 4558   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  1_oc1o 8260   ↑_m cmap 8573  Fincfn 8691   finSupp cfsupp 9058  Basecbs 16840  Scalarcsca 16891  0_gc0g 17067  Mndcmnd 18300   linC clinc 45633   LinCo clinco 45634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-map 8575  df-en 8692  df-fin 8695  df-fsupp 9059  df-seq 13650  df-0g 17069  df-gsum 17070  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-linc 45635  df-lco 45636

This theorem is referenced by:  lcoel0  45657