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Theorem mulgz 16289
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 16188 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
21adantr 465 . . 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 16288 . . 3  |-  ( ( G  e.  Mnd  /\  N  e.  NN0 )  -> 
( N  .x.  .0.  )  =  .0.  )
72, 6sylan 471 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  N  e.  NN0 )  ->  ( N  .x.  .0.  )  =  .0.  )
8 simpll 753 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  G  e.  Grp )
9 nn0z 10908 . . . . 5  |-  ( -u N  e.  NN0  ->  -u N  e.  ZZ )
109adantl 466 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  -u N  e.  ZZ )
113, 5grpidcl 16204 . . . . 5  |-  ( G  e.  Grp  ->  .0.  e.  B )
1211ad2antrr 725 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  .0.  e.  B
)
13 eqid 2457 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
143, 4, 13mulgneg 16286 . . . 4  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  .0.  e.  B )  -> 
( -u -u N  .x.  .0.  )  =  ( ( invg `  G ) `
 ( -u N  .x.  .0.  ) ) )
158, 10, 12, 14syl3anc 1228 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( -u -u N  .x.  .0.  )  =  ( ( invg `  G ) `  ( -u N  .x.  .0.  )
) )
16 zcn 10890 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
1716ad2antlr 726 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  N  e.  CC )
1817negnegd 9941 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  -u -u N  =  N )
1918oveq1d 6311 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( -u -u N  .x.  .0.  )  =  ( N  .x.  .0.  )
)
203, 4, 5mulgnn0z 16288 . . . . . 6  |-  ( ( G  e.  Mnd  /\  -u N  e.  NN0 )  ->  ( -u N  .x.  .0.  )  =  .0.  )
212, 20sylan 471 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( -u N  .x.  .0.  )  =  .0.  )
2221fveq2d 5876 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( invg `  G ) `
 ( -u N  .x.  .0.  ) )  =  ( ( invg `  G ) `  .0.  ) )
235, 13grpinvid 16227 . . . . 5  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
2423ad2antrr 725 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( invg `  G ) `
 .0.  )  =  .0.  )
2522, 24eqtrd 2498 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( invg `  G ) `
 ( -u N  .x.  .0.  ) )  =  .0.  )
2615, 19, 253eqtr3d 2506 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( N  .x.  .0.  )  =  .0.  )
27 elznn0 10900 . . . 4  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
2827simprbi 464 . . 3  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  -u N  e.  NN0 ) )
2928adantl 466 . 2  |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  e.  NN0  \/  -u N  e.  NN0 ) )
307, 26, 29mpjaodan 786 1  |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  .x.  .0.  )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   -ucneg 9825   NN0cn0 10816   ZZcz 10885   Basecbs 14643   0gc0g 14856   Mndcmnd 16045   Grpcgrp 16179   invgcminusg 16180  .gcmg 16182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12110  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-minusg 16184  df-mulg 16186
This theorem is referenced by:  odmod  16696  gexdvdsi  16729
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