Theorem isgrpde 13804

Description: Deduce a group from its properties. In this version of isgrpd 13805, 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 13684	. . 3 |- ( ( ph /\ x e. B ) -> ( x .+ .0. ) = x )
9	1, 2, 3, 4, 8	grpidd 13680	. 2 |- ( ph -> .0. = ( 0g ` G ) )
10	1, 2, 5, 6, 3, 4, 8	ismndd 13727	. 2 |- ( ph -> G e. Mnd )
11	1, 2, 9, 10, 7	isgrpd2e 13802	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   Grpcgrp 13782

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-riota 6028  df-ov 6078  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785

This theorem is referenced by:  isgrpd  13805  dfgrp2  13809  imasgrp2  13890  unitgrp  14396