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Theorem grpnnncan2 14577
Description: Cancellation law for group subtraction. (nnncan2 9100 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
grpnnncan2.b  |-  B  =  ( Base `  G
)
grpnnncan2.m  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
grpnnncan2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Z
)  .-  ( Y  .-  Z ) )  =  ( X  .-  Y
) )

Proof of Theorem grpnnncan2
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  G  e.  Grp )
2 simpr1 961 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  X  e.  B )
3 simpr3 963 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  Z  e.  B )
4 grpnnncan2.b . . . . 5  |-  B  =  ( Base `  G
)
5 grpnnncan2.m . . . . 5  |-  .-  =  ( -g `  G )
64, 5grpsubcl 14562 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  e.  B )
763adant3r1 1160 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( Y  .-  Z )  e.  B )
8 eqid 2296 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
94, 8, 5grpsubsub4 14574 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Z  e.  B  /\  ( Y  .-  Z
)  e.  B ) )  ->  ( ( X  .-  Z )  .-  ( Y  .-  Z ) )  =  ( X 
.-  ( ( Y 
.-  Z ) ( +g  `  G ) Z ) ) )
101, 2, 3, 7, 9syl13anc 1184 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Z
)  .-  ( Y  .-  Z ) )  =  ( X  .-  (
( Y  .-  Z
) ( +g  `  G
) Z ) ) )
114, 8, 5grpnpcan 14573 . . . 4  |-  ( ( G  e.  Grp  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( Y  .-  Z ) ( +g  `  G ) Z )  =  Y )
12113adant3r1 1160 . . 3  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( Y  .-  Z
) ( +g  `  G
) Z )  =  Y )
1312oveq2d 5890 . 2  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .-  ( ( Y 
.-  Z ) ( +g  `  G ) Z ) )  =  ( X  .-  Y
) )
1410, 13eqtrd 2328 1  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .-  Z
)  .-  ( Y  .-  Z ) )  =  ( X  .-  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   -gcsg 14381
This theorem is referenced by:  2idlcpbl  16002  nrmmetd  18113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507
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