Theorem isgrpi 13274

Description: Properties that determine a group. N (negative) is normally dependent on x i.e. read it as N ( x ). (Contributed by NM, 3-Sep-2011.)

Hypotheses

Ref	Expression
isgrpi.b	|- B = ( Base ` G )
isgrpi.p	|- .+ = ( +g ` G )
isgrpi.c	|- ( ( x e. B /\ y e. B ) -> ( x .+ y ) e. B )
isgrpi.a	|- ( ( x e. B /\ y e. B /\ z e. B ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
isgrpi.z	|- .0. e. B
isgrpi.i	|- ( x e. B -> ( .0. .+ x ) = x )
isgrpi.n	|- ( x e. B -> N e. B )
isgrpi.j	|- ( x e. B -> ( N .+ x ) = .0. )

Assertion

Ref	Expression
isgrpi	|- G e. Grp

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

Allowed substitution hints:   N( x, z)

Proof of Theorem isgrpi

Step	Hyp	Ref	Expression
1		isgrpi.b	. . . 4 |- B = ( Base ` G )
2	1	a1i 9	. . 3 |- ( T. -> B = ( Base ` G ) )
3		isgrpi.p	. . . 4 |- .+ = ( +g ` G )
4	3	a1i 9	. . 3 |- ( T. -> .+ = ( +g ` G ) )
5		isgrpi.c	. . . 4 |- ( ( x e. B /\ y e. B ) -> ( x .+ y ) e. B )
6	5	3adant1 1017	. . 3 |- ( ( T. /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
7		isgrpi.a	. . . 4 |- ( ( x e. B /\ y e. B /\ z e. B ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
8	7	adantl 277	. . 3 |- ( ( T. /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
9		isgrpi.z	. . . 4 |- .0. e. B
10	9	a1i 9	. . 3 |- ( T. -> .0. e. B )
11		isgrpi.i	. . . 4 |- ( x e. B -> ( .0. .+ x ) = x )
12	11	adantl 277	. . 3 |- ( ( T. /\ x e. B ) -> ( .0. .+ x ) = x )
13		isgrpi.n	. . . 4 |- ( x e. B -> N e. B )
14	13	adantl 277	. . 3 |- ( ( T. /\ x e. B ) -> N e. B )
15		isgrpi.j	. . . 4 |- ( x e. B -> ( N .+ x ) = .0. )
16	15	adantl 277	. . 3 |- ( ( T. /\ x e. B ) -> ( N .+ x ) = .0. )
17	2, 4, 6, 8, 10, 12, 14, 16	isgrpd 13273	. 2 |- ( T. -> G e. Grp )
18	17	mptru 1381	1 |- G e. Grp

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1372   T. wtru 1373   e. wcel 2175   ` cfv 5268  (class class class)co 5934   Basecbs 12751   +g cplusg 12828   Grpcgrp 13250

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-cnex 7998  ax-resscn 7999  ax-1re 8001  ax-addrcl 8004

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-iota 5229  df-fun 5270  df-fn 5271  df-fv 5276  df-riota 5889  df-ov 5937  df-inn 9019  df-2 9077  df-ndx 12754  df-slot 12755  df-base 12757  df-plusg 12841  df-0g 13008  df-mgm 13106  df-sgrp 13152  df-mnd 13167  df-grp 13253

This theorem is referenced by:  cncrng  14249