Theorem isgrpde 13325

Description: Deduce a group from its properties. In this version of isgrpd 13326, 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

Step	Hyp	Ref	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. )
8	5, 3, 4, 6, 7	grprida 13190	. . 3 |- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = x )
9	1, 2, 3, 4, 8	grpidd 13186	. 2 |- ( ph -> .0. = ( 0g ` G ) )
10	1, 2, 5, 6, 3, 4, 8	ismndd 13240	. 2 |- ( ph -> G e. Mnd )
11	1, 2, 9, 10, 7	isgrpd2e 13323	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1372   e. wcel 2175   E.wrex 2484   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880   Grpcgrp 13303

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-riota 5898  df-ov 5946  df-inn 9036  df-2 9094  df-ndx 12806  df-slot 12807  df-base 12809  df-plusg 12893  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306

This theorem is referenced by:  isgrpd  13326  dfgrp2  13330  imasgrp2  13417  unitgrp  13849