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Theorem isgrpde 18062
Description: Deduce a group from its properties. In this version of isgrpd 18063, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrpd.b (𝜑𝐵 = (Base‘𝐺))
isgrpd.p (𝜑+ = (+g𝐺))
isgrpd.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
isgrpd.a ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
isgrpd.z (𝜑0𝐵)
isgrpd.i ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
isgrpde.n ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
Assertion
Ref Expression
isgrpde (𝜑𝐺 ∈ Grp)
Distinct variable groups:   𝑥,𝑦,𝑧, +   𝑥, 0 ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧

Proof of Theorem isgrpde
StepHypRef Expression
1 isgrpd.b . 2 (𝜑𝐵 = (Base‘𝐺))
2 isgrpd.p . 2 (𝜑+ = (+g𝐺))
3 isgrpd.z . . 3 (𝜑0𝐵)
4 isgrpd.i . . 3 ((𝜑𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
5 isgrpd.c . . . 4 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
6 isgrpd.a . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
7 isgrpde.n . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
85, 3, 4, 6, 7grpridd 17873 . . 3 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
91, 2, 3, 4, 8grpidd 17869 . 2 (𝜑0 = (0g𝐺))
101, 2, 5, 6, 3, 4, 8ismndd 17921 . 2 (𝜑𝐺 ∈ Mnd)
111, 2, 9, 10, 7isgrpd2e 18060 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Grpcgrp 18041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044
This theorem is referenced by:  isgrpd  18063  dfgrp2  18066  imasgrp2  18152  unitgrp  19346
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