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Theorem mulgz 14903
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 14809 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
21adantr 452 . . 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 14902 . . 3  |-  ( ( G  e.  Mnd  /\  N  e.  NN0 )  -> 
( N  .x.  .0.  )  =  .0.  )
72, 6sylan 458 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  N  e.  NN0 )  ->  ( N  .x.  .0.  )  =  .0.  )
8 simpll 731 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  G  e.  Grp )
9 nn0z 10296 . . . . 5  |-  ( -u N  e.  NN0  ->  -u N  e.  ZZ )
109adantl 453 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  -u N  e.  ZZ )
113, 5grpidcl 14825 . . . . 5  |-  ( G  e.  Grp  ->  .0.  e.  B )
1211ad2antrr 707 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  .0.  e.  B
)
13 eqid 2435 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
143, 4, 13mulgneg 14900 . . . 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 1184 . . 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 10279 . . . . . 6  |-  ( N  e.  ZZ  ->  N  e.  CC )
1716ad2antlr 708 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  N  e.  CC )
1817negnegd 9394 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  -u -u N  =  N )
1918oveq1d 6088 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( -u -u N  .x.  .0.  )  =  ( N  .x.  .0.  )
)
203, 4, 5mulgnn0z 14902 . . . . . 6  |-  ( ( G  e.  Mnd  /\  -u N  e.  NN0 )  ->  ( -u N  .x.  .0.  )  =  .0.  )
212, 20sylan 458 . . . . 5  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( -u N  .x.  .0.  )  =  .0.  )
2221fveq2d 5724 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( inv g `  G ) `
 ( -u N  .x.  .0.  ) )  =  ( ( inv g `  G ) `  .0.  ) )
235, 13grpinvid 14848 . . . . 5  |-  ( G  e.  Grp  ->  (
( inv g `  G ) `  .0.  )  =  .0.  )
2423ad2antrr 707 . . . 4  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( inv g `  G ) `
 .0.  )  =  .0.  )
2522, 24eqtrd 2467 . . 3  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( ( inv g `  G ) `
 ( -u N  .x.  .0.  ) )  =  .0.  )
2615, 19, 253eqtr3d 2475 . 2  |-  ( ( ( G  e.  Grp  /\  N  e.  ZZ )  /\  -u N  e.  NN0 )  ->  ( N  .x.  .0.  )  =  .0.  )
27 elznn0 10288 . . . 4  |-  ( N  e.  ZZ  <->  ( N  e.  RR  /\  ( N  e.  NN0  \/  -u N  e.  NN0 ) ) )
2827simprbi 451 . . 3  |-  ( N  e.  ZZ  ->  ( N  e.  NN0  \/  -u N  e.  NN0 ) )
2928adantl 453 . 2  |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  e.  NN0  \/  -u N  e.  NN0 ) )
307, 26, 29mpjaodan 762 1  |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  .x.  .0.  )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   -ucneg 9284   NN0cn0 10213   ZZcz 10274   Basecbs 13461   0gc0g 13715   Mndcmnd 14676   Grpcgrp 14677   inv gcminusg 14678  .gcmg 14681
This theorem is referenced by:  odmod  15176  gexdvdsi  15209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-mulg 14807
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