Theorem isgrpd 13778

Description: Deduce a group from its properties. Unlike isgrpd2 13776, 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 6065	. . . . 5 |- ( y = N -> ( y .+ x ) = ( N .+ x ) )
10	9	eqeq1d 2243	. . . 4 |- ( y = N -> ( ( y .+ x ) = .0. <-> ( N .+ x ) = .0. ) )
11	10	rspcev 2923	. . 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 13777	1 |- ( ph -> G e. Grp )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   E.wrex 2523   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   Grpcgrp 13755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758

This theorem is referenced by:  isgrpi  13779  grpressid  13816  issubg2m  13942