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Theorem mulgsubdir 14849
Description: Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
mulgsubdir.b  |-  B  =  ( Base `  G
)
mulgsubdir.t  |-  .x.  =  (.g
`  G )
mulgsubdir.d  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
mulgsubdir  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  -  N )  .x.  X )  =  ( ( M  .x.  X
)  .-  ( N  .x.  X ) ) )

Proof of Theorem mulgsubdir
StepHypRef Expression
1 znegcl 10246 . . 3  |-  ( N  e.  ZZ  ->  -u N  e.  ZZ )
2 mulgsubdir.b . . . 4  |-  B  =  ( Base `  G
)
3 mulgsubdir.t . . . 4  |-  .x.  =  (.g
`  G )
4 eqid 2388 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
52, 3, 4mulgdir 14843 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  -u N  e.  ZZ  /\  X  e.  B ) )  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) ) )
61, 5syl3anr2 1237 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) ) )
7 simpr1 963 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  M  e.  ZZ )
87zcnd 10309 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  M  e.  CC )
9 simpr2 964 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  N  e.  ZZ )
109zcnd 10309 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  N  e.  CC )
118, 10negsubd 9350 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( M  +  -u N )  =  ( M  -  N
) )
1211oveq1d 6036 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  +  -u N ) 
.x.  X )  =  ( ( M  -  N )  .x.  X
) )
13 eqid 2388 . . . . . 6  |-  ( inv g `  G )  =  ( inv g `  G )
142, 3, 13mulgneg 14836 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( ( inv g `  G ) `
 ( N  .x.  X ) ) )
15143adant3r1 1162 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( -u N  .x.  X )  =  ( ( inv g `  G ) `  ( N  .x.  X ) ) )
1615oveq2d 6037 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) )  =  ( ( M  .x.  X ) ( +g  `  G ) ( ( inv g `  G
) `  ( N  .x.  X ) ) ) )
172, 3mulgcl 14835 . . . . 5  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  X  e.  B )  ->  ( M  .x.  X )  e.  B )
18173adant3r2 1163 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( M  .x.  X )  e.  B
)
192, 3mulgcl 14835 . . . . 5  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
20193adant3r1 1162 . . . 4  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( N  .x.  X )  e.  B
)
21 mulgsubdir.d . . . . 5  |-  .-  =  ( -g `  G )
222, 4, 13, 21grpsubval 14776 . . . 4  |-  ( ( ( M  .x.  X
)  e.  B  /\  ( N  .x.  X )  e.  B )  -> 
( ( M  .x.  X )  .-  ( N  .x.  X ) )  =  ( ( M 
.x.  X ) ( +g  `  G ) ( ( inv g `  G ) `  ( N  .x.  X ) ) ) )
2318, 20, 22syl2anc 643 . . 3  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X )  .-  ( N  .x.  X ) )  =  ( ( M  .x.  X ) ( +g  `  G
) ( ( inv g `  G ) `
 ( N  .x.  X ) ) ) )
2416, 23eqtr4d 2423 . 2  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  .x.  X ) ( +g  `  G ) ( -u N  .x.  X ) )  =  ( ( M  .x.  X )  .-  ( N  .x.  X ) ) )
256, 12, 243eqtr3d 2428 1  |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
)  ->  ( ( M  -  N )  .x.  X )  =  ( ( M  .x.  X
)  .-  ( N  .x.  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021    + caddc 8927    - cmin 9224   -ucneg 9225   ZZcz 10215   Basecbs 13397   +g cplusg 13457   Grpcgrp 14613   inv gcminusg 14614   -gcsg 14616  .gcmg 14617
This theorem is referenced by:  odmod  15112  odcong  15115  gexdvds  15146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-sbg 14742  df-mulg 14743
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