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Theorem grpsubadd 13004
Description: Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
Hypotheses
Ref Expression
grpsubadd.b  |-  B  =  ( Base `  G
)
grpsubadd.p  |-  .+  =  ( +g  `  G )
grpsubadd.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpsubadd  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Y
)  =  Z  <->  ( Z  .+  Y )  =  X ) )

Proof of Theorem grpsubadd
StepHypRef Expression
1 grpsubadd.b . . . . . . 7  |-  B  =  ( Base `  G
)
2 grpsubadd.p . . . . . . 7  |-  .+  =  ( +g  `  G )
3 eqid 2189 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
4 grpsubadd.m . . . . . . 7  |-  .-  =  ( -g `  G )
51, 2, 3, 4grpsubval 12962 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
653adant3 1019 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( X  .-  Y
)  =  ( X 
.+  ( ( invg `  G ) `
 Y ) ) )
76adantl 277 . . . 4  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .-  Y )  =  ( X  .+  (
( invg `  G ) `  Y
) ) )
87eqeq1d 2198 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Y
)  =  Z  <->  ( X  .+  ( ( invg `  G ) `  Y
) )  =  Z ) )
9 simpl 109 . . . 4  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  G  e.  Grp )
10 simpr1 1005 . . . . 5  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
111, 3grpinvcl 12964 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  G ) `  Y
)  e.  B )
12113ad2antr2 1165 . . . . 5  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( invg `  G ) `  Y
)  e.  B )
131, 2grpcl 12925 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B )  ->  ( X  .+  ( ( invg `  G ) `  Y
) )  e.  B
)
149, 10, 12, 13syl3anc 1249 . . . 4  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  ( ( invg `  G ) `
 Y ) )  e.  B )
15 simpr3 1007 . . . 4  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
16 simpr2 1006 . . . 4  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Y  e.  B )
171, 2grprcan 12953 . . . 4  |-  ( ( G  e.  Grp  /\  ( ( X  .+  ( ( invg `  G ) `  Y
) )  e.  B  /\  Z  e.  B  /\  Y  e.  B
) )  ->  (
( ( X  .+  ( ( invg `  G ) `  Y
) )  .+  Y
)  =  ( Z 
.+  Y )  <->  ( X  .+  ( ( invg `  G ) `  Y
) )  =  Z ) )
189, 14, 15, 16, 17syl13anc 1251 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  .+  ( ( invg `  G ) `  Y
) )  .+  Y
)  =  ( Z 
.+  Y )  <->  ( X  .+  ( ( invg `  G ) `  Y
) )  =  Z ) )
191, 2grpass 12926 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  ( ( invg `  G ) `  Y
)  e.  B  /\  Y  e.  B )
)  ->  ( ( X  .+  ( ( invg `  G ) `
 Y ) ) 
.+  Y )  =  ( X  .+  (
( ( invg `  G ) `  Y
)  .+  Y )
) )
209, 10, 12, 16, 19syl13anc 1251 . . . . 5  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  (
( invg `  G ) `  Y
) )  .+  Y
)  =  ( X 
.+  ( ( ( invg `  G
) `  Y )  .+  Y ) ) )
21 eqid 2189 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
221, 2, 21, 3grplinv 12966 . . . . . . 7  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( ( ( invg `  G ) `
 Y )  .+  Y )  =  ( 0g `  G ) )
23223ad2antr2 1165 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( invg `  G ) `  Y
)  .+  Y )  =  ( 0g `  G ) )
2423oveq2d 5907 . . . . 5  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  ( ( ( invg `  G
) `  Y )  .+  Y ) )  =  ( X  .+  ( 0g `  G ) ) )
251, 2, 21grprid 12948 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( 0g `  G ) )  =  X )
26253ad2antr1 1164 . . . . 5  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .+  ( 0g `  G ) )  =  X )
2720, 24, 263eqtrd 2226 . . . 4  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  (
( invg `  G ) `  Y
) )  .+  Y
)  =  X )
2827eqeq1d 2198 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( ( X  .+  ( ( invg `  G ) `  Y
) )  .+  Y
)  =  ( Z 
.+  Y )  <->  X  =  ( Z  .+  Y ) ) )
298, 18, 283bitr2d 216 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Y
)  =  Z  <->  X  =  ( Z  .+  Y ) ) )
30 eqcom 2191 . 2  |-  ( X  =  ( Z  .+  Y )  <->  ( Z  .+  Y )  =  X )
3129, 30bitrdi 196 1  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Y
)  =  Z  <->  ( Z  .+  Y )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   ` cfv 5231  (class class class)co 5891   Basecbs 12486   +g cplusg 12561   0gc0g 12733   Grpcgrp 12917   invgcminusg 12918   -gcsg 12919
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-inn 8939  df-2 8997  df-ndx 12489  df-slot 12490  df-base 12492  df-plusg 12574  df-0g 12735  df-mgm 12804  df-sgrp 12837  df-mnd 12850  df-grp 12920  df-minusg 12921  df-sbg 12922
This theorem is referenced by:  grpsubsub4  13009  conjghm  13182  conjnmzb  13186  ablsubadd  13218  ablsubsub23  13231
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