Theorem isgrpd 12983

Description: Deduce a group from its properties. Unlike isgrpd2 12981, this one goes straight from the base properties rather than going through Mnd. N (negative) is normally dependent on x i.e. read it as N ( x ). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 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 )
isgrpd.n	|- ( ( ph /\ x e. B ) -> N e. B )
isgrpd.j	|- ( ( ph /\ x e. B ) -> ( N .+ x ) = .0. )

Assertion

Ref	Expression
isgrpd	|- ( ph -> G e. Grp )

Distinct variable groups:   x, y, z, .+   x, .0. , y, z   x, B, y, z   y, N   ph, x, y, z   x, G, y, z

Allowed substitution hints:   N( x, z)

Proof of Theorem isgrpd

Step	Hyp	Ref	Expression
1		isgrpd.b	. 2 |- ( ph -> B = ( Base ` G ) )
2		isgrpd.p	. 2 |- ( ph -> .+ = ( +g ` G ) )
3		isgrpd.c	. 2 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
4		isgrpd.a	. 2 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
5		isgrpd.z	. 2 |- ( ph -> .0. e. B )
6		isgrpd.i	. 2 |- ( ( ph /\ x e. B ) -> ( .0. .+ x ) = x )
7		isgrpd.n	. . 3 |- ( ( ph /\ x e. B ) -> N e. B )
8		isgrpd.j	. . 3 |- ( ( ph /\ x e. B ) -> ( N .+ x ) = .0. )
9		oveq1 5904	. . . . 5 |- ( y = N -> ( y .+ x ) = ( N .+ x ) )
10	9	eqeq1d 2198	. . . 4 |- ( y = N -> ( ( y .+ x ) = .0. <-> ( N .+ x ) = .0. ) )
11	10	rspcev 2856	. . 3 |- ( ( N e. B /\ ( N .+ x ) = .0. ) -> E. y e. B ( y .+ x ) = .0. )
12	7, 8, 11	syl2anc 411	. 2 |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )
13	1, 2, 3, 4, 5, 6, 12	isgrpde 12982	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   E.wrex 2469   ` cfv 5235  (class class class)co 5897   Basecbs 12515   +g cplusg 12592   Grpcgrp 12960

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5852  df-ov 5900  df-inn 8951  df-2 9009  df-ndx 12518  df-slot 12519  df-base 12521  df-plusg 12605  df-0g 12766  df-mgm 12835  df-sgrp 12880  df-mnd 12893  df-grp 12963

This theorem is referenced by:  isgrpi  12984  grpressid  13020  issubg2m  13145