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Theorem isgrpi 14508
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
StepHypRef Expression
1 isgrpi.b . . . 4  |-  B  =  ( Base `  G
)
21a1i 10 . . 3  |-  (  T. 
->  B  =  ( Base `  G ) )
3 isgrpi.p . . . 4  |-  .+  =  ( +g  `  G )
43a1i 10 . . 3  |-  (  T. 
->  .+  =  ( +g  `  G ) )
5 isgrpi.c . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
653adant1 973 . . 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 ) ) )
87adantl 452 . . 3  |-  ( (  T.  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
9 isgrpi.z . . . 4  |-  .0.  e.  B
109a1i 10 . . 3  |-  (  T. 
->  .0.  e.  B )
11 isgrpi.i . . . 4  |-  ( x  e.  B  ->  (  .0.  .+  x )  =  x )
1211adantl 452 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
13 isgrpi.n . . . 4  |-  ( x  e.  B  ->  N  e.  B )
1413adantl 452 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  N  e.  B )
15 isgrpi.j . . . 4  |-  ( x  e.  B  ->  ( N  .+  x )  =  .0.  )
1615adantl 452 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )
172, 4, 6, 8, 10, 12, 14, 16isgrpd 14507 . 2  |-  (  T. 
->  G  e.  Grp )
1817trud 1314 1  |-  G  e. 
Grp
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362
This theorem is referenced by:  isgrpix  14509  cnaddabl  15159  cncrng  16395  grpo2grp  20901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489
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