Theorem isgrpi 13156

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 13155	. 2 |- ( T. -> G e. Grp )
18	17	mptru 1373	1 |- G e. Grp

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   T. wtru 1365   e. wcel 2167   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   Grpcgrp 13132

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135

This theorem is referenced by:  cncrng  14125