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Theorem isgrpde 18116
Description: Deduce a group from its properties. In this version of isgrpd 18117, 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 17877 . . 3 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
91, 2, 3, 4, 8grpidd 17873 . 2 (𝜑0 = (0g𝐺))
101, 2, 5, 6, 3, 4, 8ismndd 17925 . 2 (𝜑𝐺 ∈ Mnd)
111, 2, 9, 10, 7isgrpd2e 18114 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  wrex 3107  cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  Grpcgrp 18095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-riota 7093  df-ov 7138  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098
This theorem is referenced by:  isgrpd  18117  dfgrp2  18120  imasgrp2  18206  unitgrp  19413
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