MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isgrpi Structured version   Unicode version

Theorem isgrpi 16054
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 1015 . . 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 466 . . 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 466 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
13 isgrpi.n . . . 4  |-  ( x  e.  B  ->  N  e.  B )
1413adantl 466 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  N  e.  B )
15 isgrpi.j . . . 4  |-  ( x  e.  B  ->  ( N  .+  x )  =  .0.  )
1615adantl 466 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )
172, 4, 6, 8, 10, 12, 14, 16isgrpd 16053 . 2  |-  ( T. 
->  G  e.  Grp )
1817trud 1392 1  |-  G  e. 
Grp
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383   T. wtru 1384    e. wcel 1804   ` cfv 5578  (class class class)co 6281   Basecbs 14613   +g cplusg 14678   Grpcgrp 16031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-riota 6242  df-ov 6284  df-0g 14820  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-grp 16035
This theorem is referenced by:  isgrpix  16055  cnaddabl  16853  cncrng  18417  grpo2grp  25212
  Copyright terms: Public domain W3C validator