Theorem isgrpd 13887

Description: Deduce a group from its properties. Unlike isgrpd2 13885, 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 6095	. . . . 5 |- ( y = N -> ( y .+ x ) = ( N .+ x ) )
10	9	eqeq1d 2247	. . . 4 |- ( y = N -> ( ( y .+ x ) = .0. <-> ( N .+ x ) = .0. ) )
11	10	rspcev 2929	. . 3 |- ( ( N e. B /\ ( N .+ x ) = .0. ) -> E. y e. B ( y .+ x ) = .0. )
12	7, 8, 11	syl2anc 415	. 2 |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = .0. )
13	1, 2, 3, 4, 5, 6, 12	isgrpde 13886	1 |- ( ph -> G e. Grp )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   Grpcgrp 13864

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-riota 6041  df-ov 6091  df-inn 9314  df-2 9372  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867

This theorem is used by:  isgrpi  13888  grpressid  13925  issubg2m  14051