Theorem isgrpi 13737

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 1042	. . 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 13736	. 2 |- ( T. -> G e. Grp )
18	17	mptru 1407	1 |- G e. Grp

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   T. wtru 1399   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   Grpcgrp 13713

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-2 9296  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716

This theorem is referenced by:  cncrng  14717