ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  grpidrcan Unicode version

Theorem grpidrcan 12794
Description: If right adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.)
Hypotheses
Ref Expression
grpidrcan.b  |-  B  =  ( Base `  G
)
grpidrcan.p  |-  .+  =  ( +g  `  G )
grpidrcan.o  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
grpidrcan  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  =  X  <-> 
Z  =  .0.  )
)

Proof of Theorem grpidrcan
StepHypRef Expression
1 grpidrcan.b . . . . 5  |-  B  =  ( Base `  G
)
2 grpidrcan.p . . . . 5  |-  .+  =  ( +g  `  G )
3 grpidrcan.o . . . . 5  |-  .0.  =  ( 0g `  G )
41, 2, 3grprid 12767 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  .0.  )  =  X )
543adant3 1017 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .+  .0.  )  =  X )
65eqeq2d 2187 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  =  ( X  .+  .0.  )  <->  ( X  .+  Z )  =  X ) )
7 simp1 997 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  G  e.  Grp )
8 simp3 999 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  Z  e.  B )
91, 3grpidcl 12764 . . . 4  |-  ( G  e.  Grp  ->  .0.  e.  B )
1093ad2ant1 1018 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  .0.  e.  B )
11 simp2 998 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  X  e.  B )
121, 2grplcan 12791 . . 3  |-  ( ( G  e.  Grp  /\  ( Z  e.  B  /\  .0.  e.  B  /\  X  e.  B )
)  ->  ( ( X  .+  Z )  =  ( X  .+  .0.  ) 
<->  Z  =  .0.  )
)
137, 8, 10, 11, 12syl13anc 1240 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  =  ( X  .+  .0.  )  <->  Z  =  .0.  ) )
146, 13bitr3d 190 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Z  e.  B )  ->  ( ( X  .+  Z )  =  X  <-> 
Z  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2146   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   0gc0g 12626   Grpcgrp 12738
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-inn 8891  df-2 8949  df-ndx 12431  df-slot 12432  df-base 12434  df-plusg 12505  df-0g 12628  df-mgm 12640  df-sgrp 12673  df-mnd 12683  df-grp 12741  df-minusg 12742
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator