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Theorem isgrpi 14831
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 11 . . 3  |-  (  T. 
->  B  =  ( Base `  G ) )
3 isgrpi.p . . . 4  |-  .+  =  ( +g  `  G )
43a1i 11 . . 3  |-  (  T. 
->  .+  =  ( +g  `  G ) )
5 isgrpi.c . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
653adant1 975 . . 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 453 . . 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 11 . . 3  |-  (  T. 
->  .0.  e.  B )
11 isgrpi.i . . . 4  |-  ( x  e.  B  ->  (  .0.  .+  x )  =  x )
1211adantl 453 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
13 isgrpi.n . . . 4  |-  ( x  e.  B  ->  N  e.  B )
1413adantl 453 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  N  e.  B )
15 isgrpi.j . . . 4  |-  ( x  e.  B  ->  ( N  .+  x )  =  .0.  )
1615adantl 453 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )
172, 4, 6, 8, 10, 12, 14, 16isgrpd 14830 . 2  |-  (  T. 
->  G  e.  Grp )
1817trud 1332 1  |-  G  e. 
Grp
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    T. wtru 1325    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   Grpcgrp 14685
This theorem is referenced by:  isgrpix  14832  cnaddabl  15482  cncrng  16722  grpo2grp  21822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812
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