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Theorem isgrpi 14524
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 14523 . 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378
This theorem is referenced by:  isgrpix  14525  cnaddabl  15175  cncrng  16411  grpo2grp  20917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505
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