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Theorem grpidinv 12806
Description: A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (Revised by AV, 1-Sep-2021.)
Hypotheses
Ref Expression
grpidinv.b  |-  B  =  ( Base `  G
)
grpidinv.p  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
grpidinv  |-  ( G  e.  Grp  ->  E. u  e.  B  A. x  e.  B  ( (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  /\  E. y  e.  B  ( ( y  .+  x
)  =  u  /\  ( x  .+  y )  =  u ) ) )
Distinct variable groups:    u, G, x, y    u, B, y   
u,  .+ , y
Allowed substitution hints:    B( x)    .+ ( x)

Proof of Theorem grpidinv
StepHypRef Expression
1 grpidinv.b . . 3  |-  B  =  ( Base `  G
)
2 eqid 2177 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2grpidcl 12781 . 2  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
4 oveq1 5875 . . . . . . 7  |-  ( u  =  ( 0g `  G )  ->  (
u  .+  x )  =  ( ( 0g
`  G )  .+  x ) )
54eqeq1d 2186 . . . . . 6  |-  ( u  =  ( 0g `  G )  ->  (
( u  .+  x
)  =  x  <->  ( ( 0g `  G )  .+  x )  =  x ) )
6 oveq2 5876 . . . . . . 7  |-  ( u  =  ( 0g `  G )  ->  (
x  .+  u )  =  ( x  .+  ( 0g `  G ) ) )
76eqeq1d 2186 . . . . . 6  |-  ( u  =  ( 0g `  G )  ->  (
( x  .+  u
)  =  x  <->  ( x  .+  ( 0g `  G
) )  =  x ) )
85, 7anbi12d 473 . . . . 5  |-  ( u  =  ( 0g `  G )  ->  (
( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )  <->  ( ( ( 0g `  G ) 
.+  x )  =  x  /\  ( x 
.+  ( 0g `  G ) )  =  x ) ) )
9 eqeq2 2187 . . . . . . 7  |-  ( u  =  ( 0g `  G )  ->  (
( y  .+  x
)  =  u  <->  ( y  .+  x )  =  ( 0g `  G ) ) )
10 eqeq2 2187 . . . . . . 7  |-  ( u  =  ( 0g `  G )  ->  (
( x  .+  y
)  =  u  <->  ( x  .+  y )  =  ( 0g `  G ) ) )
119, 10anbi12d 473 . . . . . 6  |-  ( u  =  ( 0g `  G )  ->  (
( ( y  .+  x )  =  u  /\  ( x  .+  y )  =  u )  <->  ( ( y 
.+  x )  =  ( 0g `  G
)  /\  ( x  .+  y )  =  ( 0g `  G ) ) ) )
1211rexbidv 2478 . . . . 5  |-  ( u  =  ( 0g `  G )  ->  ( E. y  e.  B  ( ( y  .+  x )  =  u  /\  ( x  .+  y )  =  u )  <->  E. y  e.  B  ( ( y  .+  x )  =  ( 0g `  G )  /\  ( x  .+  y )  =  ( 0g `  G ) ) ) )
138, 12anbi12d 473 . . . 4  |-  ( u  =  ( 0g `  G )  ->  (
( ( ( u 
.+  x )  =  x  /\  ( x 
.+  u )  =  x )  /\  E. y  e.  B  (
( y  .+  x
)  =  u  /\  ( x  .+  y )  =  u ) )  <-> 
( ( ( ( 0g `  G ) 
.+  x )  =  x  /\  ( x 
.+  ( 0g `  G ) )  =  x )  /\  E. y  e.  B  (
( y  .+  x
)  =  ( 0g
`  G )  /\  ( x  .+  y )  =  ( 0g `  G ) ) ) ) )
1413ralbidv 2477 . . 3  |-  ( u  =  ( 0g `  G )  ->  ( A. x  e.  B  ( ( ( u 
.+  x )  =  x  /\  ( x 
.+  u )  =  x )  /\  E. y  e.  B  (
( y  .+  x
)  =  u  /\  ( x  .+  y )  =  u ) )  <->  A. x  e.  B  ( ( ( ( 0g `  G ) 
.+  x )  =  x  /\  ( x 
.+  ( 0g `  G ) )  =  x )  /\  E. y  e.  B  (
( y  .+  x
)  =  ( 0g
`  G )  /\  ( x  .+  y )  =  ( 0g `  G ) ) ) ) )
1514adantl 277 . 2  |-  ( ( G  e.  Grp  /\  u  =  ( 0g `  G ) )  -> 
( A. x  e.  B  ( ( ( u  .+  x )  =  x  /\  (
x  .+  u )  =  x )  /\  E. y  e.  B  (
( y  .+  x
)  =  u  /\  ( x  .+  y )  =  u ) )  <->  A. x  e.  B  ( ( ( ( 0g `  G ) 
.+  x )  =  x  /\  ( x 
.+  ( 0g `  G ) )  =  x )  /\  E. y  e.  B  (
( y  .+  x
)  =  ( 0g
`  G )  /\  ( x  .+  y )  =  ( 0g `  G ) ) ) ) )
16 grpidinv.p . . . 4  |-  .+  =  ( +g  `  G )
171, 16, 2grpidinv2 12805 . . 3  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( ( ( ( 0g `  G ) 
.+  x )  =  x  /\  ( x 
.+  ( 0g `  G ) )  =  x )  /\  E. y  e.  B  (
( y  .+  x
)  =  ( 0g
`  G )  /\  ( x  .+  y )  =  ( 0g `  G ) ) ) )
1817ralrimiva 2550 . 2  |-  ( G  e.  Grp  ->  A. x  e.  B  ( (
( ( 0g `  G )  .+  x
)  =  x  /\  ( x  .+  ( 0g
`  G ) )  =  x )  /\  E. y  e.  B  ( ( y  .+  x
)  =  ( 0g
`  G )  /\  ( x  .+  y )  =  ( 0g `  G ) ) ) )
193, 15, 18rspcedvd 2847 1  |-  ( G  e.  Grp  ->  E. u  e.  B  A. x  e.  B  ( (
( u  .+  x
)  =  x  /\  ( x  .+  u )  =  x )  /\  E. y  e.  B  ( ( y  .+  x
)  =  u  /\  ( x  .+  y )  =  u ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640   Grpcgrp 12754
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-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
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-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758
This theorem is referenced by: (None)
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