Theorem grp1 13238

Description: The (smallest) structure representing a trivial group. According to Wikipedia ("Trivial group", 28-Apr-2019, https://en.wikipedia.org/wiki/Trivial_group) "In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element". (Contributed by AV, 28-Apr-2019.)

Hypothesis

Ref	Expression
grp1.m	⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}

Assertion

Ref	Expression
grp1	⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp)

Proof of Theorem grp1

Dummy variables 𝑒 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grp1.m	. . 3 ⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
2	1	mnd1 13087	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)
3		df-ov 5925	. . . . 5 ⊢ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩}‘⟨𝐼, 𝐼⟩)
4		opexg 4261	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ⟨𝐼, 𝐼⟩ ∈ V)
5	4	anidms 397	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ⟨𝐼, 𝐼⟩ ∈ V)
6		fvsng 5758	. . . . . 6 ⊢ ((⟨𝐼, 𝐼⟩ ∈ V ∧ 𝐼 ∈ 𝑉) → ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩}‘⟨𝐼, 𝐼⟩) = 𝐼)
7	5, 6	mpancom 422	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩}‘⟨𝐼, 𝐼⟩) = 𝐼)
8	3, 7	eqtrid 2241	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = 𝐼)
9	1	mnd1id 13088	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑀) = 𝐼)
10	8, 9	eqtr4d 2232	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀))
11		oveq2 5930	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼))
12	11	eqeq1d 2205	. . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
13	12	rexbidv 2498	. . . . 5 ⊢ (𝑖 = 𝐼 → (∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ ∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
14	13	ralsng 3662	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ ∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
15		oveq1 5929	. . . . . 6 ⊢ (𝑒 = 𝐼 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼))
16	15	eqeq1d 2205	. . . . 5 ⊢ (𝑒 = 𝐼 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀) ↔ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
17	16	rexsng 3663	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀) ↔ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
18	14, 17	bitrd 188	. . 3 ⊢ (𝐼 ∈ 𝑉 → (∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
19	10, 18	mpbird 167	. 2 ⊢ (𝐼 ∈ 𝑉 → ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀))
20		eqid 2196	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
21		eqid 2196	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
22		eqid 2196	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
23	20, 21, 22	isgrp 13138	. . 3 ⊢ (𝑀 ∈ Grp ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ (Base‘𝑀)∃𝑒 ∈ (Base‘𝑀)(𝑒(+g‘𝑀)𝑖) = (0g‘𝑀)))
24		snexg 4217	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {𝐼} ∈ V)
25		opexg 4261	. . . . . . . 8 ⊢ ((⟨𝐼, 𝐼⟩ ∈ V ∧ 𝐼 ∈ 𝑉) → ⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V)
26	5, 25	mpancom 422	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V)
27		snexg 4217	. . . . . . 7 ⊢ (⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V)
28	26, 27	syl 14	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V)
29	1	grpbaseg 12804	. . . . . 6 ⊢ (({𝐼} ∈ V ∧ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V) → {𝐼} = (Base‘𝑀))
30	24, 28, 29	syl2anc 411	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → {𝐼} = (Base‘𝑀))
31	1	grpplusgg 12805	. . . . . . . . 9 ⊢ (({𝐼} ∈ V ∧ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V) → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} = (+g‘𝑀))
32	24, 28, 31	syl2anc 411	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} = (+g‘𝑀))
33	32	oveqd 5939	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (𝑒(+g‘𝑀)𝑖))
34	33	eqeq1d 2205	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ (𝑒(+g‘𝑀)𝑖) = (0g‘𝑀)))
35	30, 34	rexeqbidv 2710	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ ∃𝑒 ∈ (Base‘𝑀)(𝑒(+g‘𝑀)𝑖) = (0g‘𝑀)))
36	30, 35	raleqbidv 2709	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ ∀𝑖 ∈ (Base‘𝑀)∃𝑒 ∈ (Base‘𝑀)(𝑒(+g‘𝑀)𝑖) = (0g‘𝑀)))
37	36	anbi2d 464	. . 3 ⊢ (𝐼 ∈ 𝑉 → ((𝑀 ∈ Mnd ∧ ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀)) ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ (Base‘𝑀)∃𝑒 ∈ (Base‘𝑀)(𝑒(+g‘𝑀)𝑖) = (0g‘𝑀))))
38	23, 37	bitr4id 199	. 2 ⊢ (𝐼 ∈ 𝑉 → (𝑀 ∈ Grp ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀))))
39	2, 19, 38	mpbir2and 946	1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  Vcvv 2763  {csn 3622  {cpr 3623  ⟨cop 3625  ‘cfv 5258  (class class class)co 5922  ndxcnx 12675  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927  Mndcmnd 13057  Grpcgrp 13132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135

This theorem is referenced by:  grp1inv  13239  ring1  13615