Theorem grp1 13513

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 13362	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd)
3		df-ov 5960	. . . . 5 ⊢ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩}‘⟨𝐼, 𝐼⟩)
4		opexg 4280	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ∈ 𝑉) → ⟨𝐼, 𝐼⟩ ∈ V)
5	4	anidms 397	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ⟨𝐼, 𝐼⟩ ∈ V)
6		fvsng 5793	. . . . . 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 2251	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = 𝐼)
9	1	mnd1id 13363	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑀) = 𝐼)
10	8, 9	eqtr4d 2242	. . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀))
11		oveq2 5965	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼))
12	11	eqeq1d 2215	. . . . . 6 ⊢ (𝑖 = 𝐼 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
13	12	rexbidv 2508	. . . . 5 ⊢ (𝑖 = 𝐼 → (∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ ∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
14	13	ralsng 3678	. . . 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 5964	. . . . . 6 ⊢ (𝑒 = 𝐼 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼))
16	15	eqeq1d 2215	. . . . 5 ⊢ (𝑒 = 𝐼 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀) ↔ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g‘𝑀)))
17	16	rexsng 3679	. . . 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 2206	. . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀)
21		eqid 2206	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑀)
22		eqid 2206	. . . 4 ⊢ (0g‘𝑀) = (0g‘𝑀)
23	20, 21, 22	isgrp 13413	. . 3 ⊢ (𝑀 ∈ Grp ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ (Base‘𝑀)∃𝑒 ∈ (Base‘𝑀)(𝑒(+g‘𝑀)𝑖) = (0g‘𝑀)))
24		snexg 4236	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {𝐼} ∈ V)
25		opexg 4280	. . . . . . . 8 ⊢ ((⟨𝐼, 𝐼⟩ ∈ V ∧ 𝐼 ∈ 𝑉) → ⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V)
26	5, 25	mpancom 422	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V)
27		snexg 4236	. . . . . . 7 ⊢ (⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V)
28	26, 27	syl 14	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V)
29	1	grpbaseg 13034	. . . . . 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 13035	. . . . . . . . 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 5974	. . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (𝑒(+g‘𝑀)𝑖))
34	33	eqeq1d 2215	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g‘𝑀) ↔ (𝑒(+g‘𝑀)𝑖) = (0g‘𝑀)))
35	30, 34	rexeqbidv 2720	. . . . 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 2719	. . . 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 947	1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  Vcvv 2773  {csn 3638  {cpr 3639  ⟨cop 3641  ‘cfv 5280  (class class class)co 5957  ndxcnx 12904  Basecbs 12907  +_gcplusg 12984  0_gc0g 13163  Mndcmnd 13323  Grpcgrp 13407

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288  df-riota 5912  df-ov 5960  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-ndx 12910  df-slot 12911  df-base 12913  df-plusg 12997  df-0g 13165  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410

This theorem is referenced by:  grp1inv  13514  ring1  13896