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Theorem dfgrp2e 12793
Description: Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.)
Hypotheses
Ref Expression
dfgrp2.b  |-  B  =  ( Base `  G
)
dfgrp2.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
dfgrp2e  |-  ( G  e.  Grp  <->  ( A. x  e.  B  A. y  e.  B  (
( x  .+  y
)  e.  B  /\  A. z  e.  B  ( ( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )  /\  E. n  e.  B  A. x  e.  B  (
( n  .+  x
)  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n ) ) )
Distinct variable groups:    B, i, n, x    i, G, n, x    .+ , i, n, x   
y, B, z, x   
y, G, z    y,  .+ , z

Proof of Theorem dfgrp2e
StepHypRef Expression
1 dfgrp2.b . . 3  |-  B  =  ( Base `  G
)
2 dfgrp2.p . . 3  |-  .+  =  ( +g  `  G )
31, 2dfgrp2 12792 . 2  |-  ( G  e.  Grp  <->  ( G  e. Smgrp  /\  E. n  e.  B  A. x  e.  B  ( ( n 
.+  x )  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n ) ) )
4 rexm 3522 . . . 4  |-  ( E. n  e.  B  A. x  e.  B  (
( n  .+  x
)  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n )  ->  E. n  n  e.  B )
51basmex 12503 . . . . 5  |-  ( n  e.  B  ->  G  e.  _V )
65exlimiv 1598 . . . 4  |-  ( E. n  n  e.  B  ->  G  e.  _V )
71, 2issgrpv 12702 . . . 4  |-  ( G  e.  _V  ->  ( G  e. Smgrp  <->  A. x  e.  B  A. y  e.  B  ( ( x  .+  y )  e.  B  /\  A. z  e.  B  ( ( x  .+  y )  .+  z
)  =  ( x 
.+  ( y  .+  z ) ) ) ) )
84, 6, 73syl 17 . . 3  |-  ( E. n  e.  B  A. x  e.  B  (
( n  .+  x
)  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n )  -> 
( G  e. Smgrp  <->  A. x  e.  B  A. y  e.  B  ( (
x  .+  y )  e.  B  /\  A. z  e.  B  ( (
x  .+  y )  .+  z )  =  ( x  .+  ( y 
.+  z ) ) ) ) )
98pm5.32ri 455 . 2  |-  ( ( G  e. Smgrp  /\  E. n  e.  B  A. x  e.  B  ( (
n  .+  x )  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n ) )  <->  ( A. x  e.  B  A. y  e.  B  (
( x  .+  y
)  e.  B  /\  A. z  e.  B  ( ( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )  /\  E. n  e.  B  A. x  e.  B  (
( n  .+  x
)  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n ) ) )
103, 9bitri 184 1  |-  ( G  e.  Grp  <->  ( A. x  e.  B  A. y  e.  B  (
( x  .+  y
)  e.  B  /\  A. z  e.  B  ( ( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )  /\  E. n  e.  B  A. x  e.  B  (
( n  .+  x
)  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2737   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518  Smgrpcsgrp 12699   Grpcgrp 12767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7893  ax-resscn 7894  ax-1re 7896  ax-addrcl 7899
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8909  df-2 8967  df-ndx 12448  df-slot 12449  df-base 12451  df-plusg 12531  df-0g 12655  df-mgm 12667  df-sgrp 12700  df-mnd 12710  df-grp 12770
This theorem is referenced by: (None)
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