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Theorem isgrpde 18879
Description: Deduce a group from its properties. In this version of isgrpd 18880, 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, 7grprida 18600 . . 3 ((𝜑𝑥𝐵) → (𝑥 + 0 ) = 𝑥)
91, 2, 3, 4, 8grpidd 18596 . 2 (𝜑0 = (0g𝐺))
101, 2, 5, 6, 3, 4, 8ismndd 18681 . 2 (𝜑𝐺 ∈ Mnd)
111, 2, 9, 10, 7isgrpd2e 18877 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  wrex 3068  cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  Grpcgrp 18855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7367  df-ov 7414  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858
This theorem is referenced by:  isgrpd  18880  dfgrp2  18883  imasgrp2  18974  unitgrp  20274
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