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Theorem isgrpde 16053
Description: Deduce a group from its properties. In this version of isgrpd 16054, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrpd.b  |-  ( ph  ->  B  =  ( Base `  G ) )
isgrpd.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
isgrpd.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
isgrpd.a  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
isgrpd.z  |-  ( ph  ->  .0.  e.  B )
isgrpd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
isgrpde.n  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
Assertion
Ref Expression
isgrpde  |-  ( ph  ->  G  e.  Grp )
Distinct variable groups:    x, y,
z,  .+    x,  .0. , y,
z    x, B, y, z    ph, x, y, z    x, G, y, z

Proof of Theorem isgrpde
StepHypRef Expression
1 isgrpd.b . 2  |-  ( ph  ->  B  =  ( Base `  G ) )
2 isgrpd.p . 2  |-  ( ph  ->  .+  =  ( +g  `  G ) )
3 isgrpd.z . . 3  |-  ( ph  ->  .0.  e.  B )
4 isgrpd.i . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
5 isgrpd.c . . . 4  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
6 isgrpd.a . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
7 isgrpde.n . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
85, 3, 4, 6, 7grpridd 6500 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
91, 2, 3, 4, 8grpidd 15874 . 2  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
101, 2, 5, 6, 3, 4, 8ismndd 15922 . 2  |-  ( ph  ->  G  e.  Mnd )
111, 2, 9, 10, 7isgrpd2e 16051 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   E.wrex 2794   ` cfv 5578  (class class class)co 6281   Basecbs 14614   +g cplusg 14679   Grpcgrp 16032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-riota 6242  df-ov 6284  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036
This theorem is referenced by:  isgrpd  16054  imasgrp2  16164  unitgrp  17295
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