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Theorem mulgz 14604
Description: A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
Hypotheses
Ref Expression
mulgnn0z.b  |-  B  =  ( Base `  G
)
mulgnn0z.t  |-  .x.  =  (.g
`  G )
mulgnn0z.o  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
mulgz  |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  .x.  .0.  )  =  .0.  )

Proof of Theorem mulgz
StepHypRef Expression
1 grpmnd 14510 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
21adantr 451 . . 3  |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  G  e.  Mnd )
3 mulgnn0z.b . . . 4  |-  B  =  ( Base `  G
)
4 mulgnn0z.t . . . 4  |-  .x.  =  (.g
`  G )
5 mulgnn0z.o . . . 4  |-  .0.  =  ( 0g `  G )
63, 4, 5mulgnn0z 14603 . . 3  |-  ( ( G  e.  Mnd  /\  N  e.  NN0 )  -> 
( N  .x.  .0.  )  =  .0.  )
72, 6sylan 457 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  N  e.  NN0 )  ->  ( N  .x.  .0.  )  =  .0.  )
8 simpll 730 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  G  e.  Grp )
9 nn0z 10062 . . . . 5  |-  ( -u N  e.  NN0  ->  -u N  e.  ZZ )
109adantl 452 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  -u N  e.  ZZ )
113, 5grpidcl 14526 . . . . 5  |-  ( G  e.  Grp  ->  .0.  e.  B )
1211ad2antrr 706 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  .0.  e.  B
)
13 eqid 2296 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
143, 4, 13mulgneg 14601 . . . 4  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  .0.  e.  B )  -> 
( -u -u N  .x.  .0.  )  =  ( ( inv g `  G ) `
 ( -u N  .x.  .0.  ) ) )
158, 10, 12, 14syl3anc 1182 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( -u -u N  .x.  .0.  )  =  ( ( inv g `  G ) `  ( -u N  .x.  .0.  )
) )
16 zcn 10045 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
1716ad2antlr 707 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  N  e.  CC )
1817negnegd 9164 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  -u -u N  =  N )
1918oveq1d 5889 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( -u -u N  .x.  .0.  )  =  ( N  .x.  .0.  )
)
203, 4, 5mulgnn0z 14603 . . . . . 6  |-  ( ( G  e.  Mnd  /\  -u N  e.  NN0 )  ->  ( -u N  .x.  .0.  )  =  .0.  )
212, 20sylan 457 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( -u N  .x.  .0.  )  =  .0.  )
2221fveq2d 5545 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( inv g `  G ) `
 ( -u N  .x.  .0.  ) )  =  ( ( inv g `  G ) `  .0.  ) )
235, 13grpinvid 14549 . . . . 5  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
2423ad2antrr 706 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( inv g `  G ) `
 .0.  )  =  .0.  )
2522, 24eqtrd 2328 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( inv g `  G ) `
 ( -u N  .x.  .0.  ) )  =  .0.  )
2615, 19, 253eqtr3d 2336 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( N  .x.  .0.  )  =  .0.  )
27 elznn0 10054 . . . 4  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
2827simprbi 450 . . 3  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  -u N  e.  NN0 ) )
2928adantl 452 . 2  |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  e.  NN0  \/  -u N  e.  NN0 ) )
307, 26, 29mpjaodan 761 1  |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  .x.  .0.  )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   -ucneg 9054   NN0cn0 9981   ZZcz 10040   Basecbs 13164   0gc0g 13416   Mndcmnd 14377   Grpcgrp 14378   inv gcminusg 14379  .gcmg 14382
This theorem is referenced by:  odmod  14877  gexdvdsi  14910
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  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mulg 14508
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