Theorem isgrpi 12905

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 1015	. . 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 12904	. 2 |- ( T. -> G e. Grp )
18	17	mptru 1362	1 |- G e. Grp

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   T. wtru 1354   e. wcel 2148   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   Grpcgrp 12882

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885

This theorem is referenced by:  cncrng  13548