Theorem isgrpi 13552

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 1039	. . 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 13551	. 2 |- ( T. -> G e. Grp )
18	17	mptru 1404	1 |- G e. Grp

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   T. wtru 1396   e. wcel 2200   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   Grpcgrp 13528

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-inn 9107  df-2 9165  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531

This theorem is referenced by:  cncrng  14527