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Theorem isgrpde 15868
Description: Deduce a group from its properties. In this version of isgrpd 15869, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 6-Jan-2015.)
Hypotheses
Ref Expression
isgrpd.b  |-  ( ph  ->  B  =  ( Base `  G ) )
isgrpd.p  |-  ( ph  ->  .+  =  ( +g  `  G ) )
isgrpd.c  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
isgrpd.a  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
isgrpd.z  |-  ( ph  ->  .0.  e.  B )
isgrpd.i  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
isgrpde.n  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
Assertion
Ref Expression
isgrpde  |-  ( ph  ->  G  e.  Grp )
Distinct variable groups:    x, y,
z,  .+    x,  .0. , y,
z    x, B, y, z    ph, x, y, z    x, G, y, z

Proof of Theorem isgrpde
StepHypRef Expression
1 isgrpd.b . 2  |-  ( ph  ->  B  =  ( Base `  G ) )
2 isgrpd.p . 2  |-  ( ph  ->  .+  =  ( +g  `  G ) )
3 isgrpd.z . . 3  |-  ( ph  ->  .0.  e.  B )
4 isgrpd.i . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )
5 isgrpd.c . . . 4  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .+  y )  e.  B
)
6 isgrpd.a . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) )
7 isgrpde.n . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )
85, 3, 4, 6, 7grpridd 6490 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  (
x  .+  .0.  )  =  x )
91, 2, 3, 4, 8grpidd 15749 . 2  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
101, 2, 5, 6, 3, 4, 8ismndd 15750 . 2  |-  ( ph  ->  G  e.  Mnd )
111, 2, 9, 10, 7isgrpd2e 15866 1  |-  ( ph  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   E.wrex 2808   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-riota 6236  df-ov 6278  df-0g 14686  df-mnd 15721  df-grp 15851
This theorem is referenced by:  isgrpd  15869  imasgrp2  15978  unitgrp  17093
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