Theorem isgrpde 13097

Description: Deduce a group from its properties. In this version of isgrpd 13098, 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 12973	. . 3 |- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = x )
9	1, 2, 3, 4, 8	grpidd 12969	. 2 |- ( ph -> .0. = ( 0g ` G ) )
10	1, 2, 5, 6, 3, 4, 8	ismndd 13021	. 2 |- ( ph -> G e. Mnd )
11	1, 2, 9, 10, 7	isgrpd2e 13095	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   Grpcgrp 13075

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078

This theorem is referenced by:  isgrpd  13098  dfgrp2  13102  imasgrp2  13183  unitgrp  13615