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Theorem grp1 12852
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 (𝐼𝑉𝑀 ∈ Grp)

Proof of Theorem grp1
Dummy variables 𝑒 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grp1.m . . 3 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}
21mnd1 12724 . 2 (𝐼𝑉𝑀 ∈ Mnd)
3 df-ov 5871 . . . . 5 (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩}‘⟨𝐼, 𝐼⟩)
4 opexg 4224 . . . . . . 7 ((𝐼𝑉𝐼𝑉) → ⟨𝐼, 𝐼⟩ ∈ V)
54anidms 397 . . . . . 6 (𝐼𝑉 → ⟨𝐼, 𝐼⟩ ∈ V)
6 fvsng 5707 . . . . . 6 ((⟨𝐼, 𝐼⟩ ∈ V ∧ 𝐼𝑉) → ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩}‘⟨𝐼, 𝐼⟩) = 𝐼)
75, 6mpancom 422 . . . . 5 (𝐼𝑉 → ({⟨⟨𝐼, 𝐼⟩, 𝐼⟩}‘⟨𝐼, 𝐼⟩) = 𝐼)
83, 7eqtrid 2222 . . . 4 (𝐼𝑉 → (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = 𝐼)
91mnd1id 12725 . . . 4 (𝐼𝑉 → (0g𝑀) = 𝐼)
108, 9eqtr4d 2213 . . 3 (𝐼𝑉 → (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀))
11 oveq2 5876 . . . . . . 7 (𝑖 = 𝐼 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼))
1211eqeq1d 2186 . . . . . 6 (𝑖 = 𝐼 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀) ↔ (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀)))
1312rexbidv 2478 . . . . 5 (𝑖 = 𝐼 → (∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀) ↔ ∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀)))
1413ralsng 3631 . . . 4 (𝐼𝑉 → (∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀) ↔ ∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀)))
15 oveq1 5875 . . . . . 6 (𝑒 = 𝐼 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼))
1615eqeq1d 2186 . . . . 5 (𝑒 = 𝐼 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀) ↔ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀)))
1716rexsng 3632 . . . 4 (𝐼𝑉 → (∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀) ↔ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀)))
1814, 17bitrd 188 . . 3 (𝐼𝑉 → (∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀) ↔ (𝐼{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝐼) = (0g𝑀)))
1910, 18mpbird 167 . 2 (𝐼𝑉 → ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀))
20 eqid 2177 . . . 4 (Base‘𝑀) = (Base‘𝑀)
21 eqid 2177 . . . 4 (+g𝑀) = (+g𝑀)
22 eqid 2177 . . . 4 (0g𝑀) = (0g𝑀)
2320, 21, 22isgrp 12760 . . 3 (𝑀 ∈ Grp ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ (Base‘𝑀)∃𝑒 ∈ (Base‘𝑀)(𝑒(+g𝑀)𝑖) = (0g𝑀)))
24 snexg 4181 . . . . . 6 (𝐼𝑉 → {𝐼} ∈ V)
25 opexg 4224 . . . . . . . 8 ((⟨𝐼, 𝐼⟩ ∈ V ∧ 𝐼𝑉) → ⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V)
265, 25mpancom 422 . . . . . . 7 (𝐼𝑉 → ⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V)
27 snexg 4181 . . . . . . 7 (⟨⟨𝐼, 𝐼⟩, 𝐼⟩ ∈ V → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V)
2826, 27syl 14 . . . . . 6 (𝐼𝑉 → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V)
291grpbaseg 12551 . . . . . 6 (({𝐼} ∈ V ∧ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V) → {𝐼} = (Base‘𝑀))
3024, 28, 29syl2anc 411 . . . . 5 (𝐼𝑉 → {𝐼} = (Base‘𝑀))
311grpplusgg 12552 . . . . . . . . 9 (({𝐼} ∈ V ∧ {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} ∈ V) → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} = (+g𝑀))
3224, 28, 31syl2anc 411 . . . . . . . 8 (𝐼𝑉 → {⟨⟨𝐼, 𝐼⟩, 𝐼⟩} = (+g𝑀))
3332oveqd 5885 . . . . . . 7 (𝐼𝑉 → (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (𝑒(+g𝑀)𝑖))
3433eqeq1d 2186 . . . . . 6 (𝐼𝑉 → ((𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀) ↔ (𝑒(+g𝑀)𝑖) = (0g𝑀)))
3530, 34rexeqbidv 2685 . . . . 5 (𝐼𝑉 → (∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀) ↔ ∃𝑒 ∈ (Base‘𝑀)(𝑒(+g𝑀)𝑖) = (0g𝑀)))
3630, 35raleqbidv 2684 . . . 4 (𝐼𝑉 → (∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀) ↔ ∀𝑖 ∈ (Base‘𝑀)∃𝑒 ∈ (Base‘𝑀)(𝑒(+g𝑀)𝑖) = (0g𝑀)))
3736anbi2d 464 . . 3 (𝐼𝑉 → ((𝑀 ∈ Mnd ∧ ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀)) ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ (Base‘𝑀)∃𝑒 ∈ (Base‘𝑀)(𝑒(+g𝑀)𝑖) = (0g𝑀))))
3823, 37bitr4id 199 . 2 (𝐼𝑉 → (𝑀 ∈ Grp ↔ (𝑀 ∈ Mnd ∧ ∀𝑖 ∈ {𝐼}∃𝑒 ∈ {𝐼} (𝑒{⟨⟨𝐼, 𝐼⟩, 𝐼⟩}𝑖) = (0g𝑀))))
392, 19, 38mpbir2and 944 1 (𝐼𝑉𝑀 ∈ Grp)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  wrex 2456  Vcvv 2737  {csn 3591  {cpr 3592  cop 3594  cfv 5211  (class class class)co 5868  ndxcnx 12429  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Mndcmnd 12696  Grpcgrp 12754
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757
This theorem is referenced by:  grp1inv  12853  ring1  13049
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