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Theorem ghmmulg 15049
Description: A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
ghmmulg.b  |-  B  =  ( Base `  G
)
ghmmulg.s  |-  .x.  =  (.g
`  G )
ghmmulg.t  |-  .X.  =  (.g
`  H )
Assertion
Ref Expression
ghmmulg  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )

Proof of Theorem ghmmulg
StepHypRef Expression
1 ghmmhm 15047 . . . . . 6  |-  ( F  e.  ( G  GrpHom  H )  ->  F  e.  ( G MndHom  H ) )
2 ghmmulg.b . . . . . . 7  |-  B  =  ( Base `  G
)
3 ghmmulg.s . . . . . . 7  |-  .x.  =  (.g
`  G )
4 ghmmulg.t . . . . . . 7  |-  .X.  =  (.g
`  H )
52, 3, 4mhmmulg 14953 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
61, 5syl3an1 1218 . . . . 5  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
763expa 1154 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  NN0 )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
87an32s 781 . . 3  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( F `  ( N  .x.  X ) )  =  ( N 
.X.  ( F `  X ) ) )
983adantl2 1115 . 2  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
10 simpl1 961 . . . . . . . 8  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  F  e.  ( G  GrpHom  H ) )
1110, 1syl 16 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  F  e.  ( G MndHom  H ) )
12 nnnn0 10259 . . . . . . . 8  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
1312ad2antll 711 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  NN0 )
14 simpl3 963 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  X  e.  B )
152, 3, 4mhmmulg 14953 . . . . . . 7  |-  ( ( F  e.  ( G MndHom  H )  /\  -u N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( -u N  .x.  X ) )  =  ( -u N  .X.  ( F `  X ) ) )
1611, 13, 14, 15syl3anc 1185 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  ( -u N  .x.  X ) )  =  ( -u N  .X.  ( F `  X ) ) )
1716fveq2d 5761 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( inv g `  H ) `  ( F `  ( -u N  .x.  X ) ) )  =  ( ( inv g `  H ) `
 ( -u N  .X.  ( F `  X
) ) ) )
18 ghmgrp1 15039 . . . . . . . 8  |-  ( F  e.  ( G  GrpHom  H )  ->  G  e.  Grp )
1910, 18syl 16 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  G  e.  Grp )
20 nnz 10334 . . . . . . . 8  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
2120ad2antll 711 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u N  e.  ZZ )
222, 3mulgcl 14938 . . . . . . 7  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
2319, 21, 14, 22syl3anc 1185 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u N  .x.  X
)  e.  B )
24 eqid 2442 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
25 eqid 2442 . . . . . . 7  |-  ( inv g `  H )  =  ( inv g `  H )
262, 24, 25ghminv 15044 . . . . . 6  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  ( -u N  .x.  X )  e.  B )  -> 
( F `  (
( inv g `  G ) `  ( -u N  .x.  X ) ) )  =  ( ( inv g `  H ) `  ( F `  ( -u N  .x.  X ) ) ) )
2710, 23, 26syl2anc 644 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  (
( inv g `  G ) `  ( -u N  .x.  X ) ) )  =  ( ( inv g `  H ) `  ( F `  ( -u N  .x.  X ) ) ) )
28 ghmgrp2 15040 . . . . . . 7  |-  ( F  e.  ( G  GrpHom  H )  ->  H  e.  Grp )
2910, 28syl 16 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  H  e.  Grp )
30 eqid 2442 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
312, 30ghmf 15041 . . . . . . . 8  |-  ( F  e.  ( G  GrpHom  H )  ->  F : B
--> ( Base `  H
) )
3210, 31syl 16 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  F : B --> ( Base `  H ) )
3332, 14ffvelrnd 5900 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  X
)  e.  ( Base `  H ) )
3430, 4, 25mulgneg 14939 . . . . . 6  |-  ( ( H  e.  Grp  /\  -u N  e.  ZZ  /\  ( F `  X )  e.  ( Base `  H
) )  ->  ( -u -u N  .X.  ( F `
 X ) )  =  ( ( inv g `  H ) `
 ( -u N  .X.  ( F `  X
) ) ) )
3529, 21, 33, 34syl3anc 1185 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u -u N  .X.  ( F `  X )
)  =  ( ( inv g `  H
) `  ( -u N  .X.  ( F `  X
) ) ) )
3617, 27, 353eqtr4d 2484 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  (
( inv g `  G ) `  ( -u N  .x.  X ) ) )  =  (
-u -u N  .X.  ( F `  X )
) )
372, 3, 24mulgneg 14939 . . . . . . 7  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( ( inv g `  G ) `  ( -u N  .x.  X ) ) )
3819, 21, 14, 37syl3anc 1185 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u -u N  .x.  X
)  =  ( ( inv g `  G
) `  ( -u N  .x.  X ) ) )
39 simprl 734 . . . . . . . . 9  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4039recnd 9145 . . . . . . . 8  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4140negnegd 9433 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N
)
4241oveq1d 6125 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u -u N  .x.  X
)  =  ( N 
.x.  X ) )
4338, 42eqtr3d 2476 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( inv g `  G ) `  ( -u N  .x.  X ) )  =  ( N 
.x.  X ) )
4443fveq2d 5761 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  (
( inv g `  G ) `  ( -u N  .x.  X ) ) )  =  ( F `  ( N 
.x.  X ) ) )
4536, 44eqtr3d 2476 . . 3  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u -u N  .X.  ( F `  X )
)  =  ( F `
 ( N  .x.  X ) ) )
4641oveq1d 6125 . . 3  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u -u N  .X.  ( F `  X )
)  =  ( N 
.X.  ( F `  X ) ) )
4745, 46eqtr3d 2476 . 2  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
48 simp2 959 . . 3  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
49 elznn0nn 10326 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
5048, 49sylib 190 . 2  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
519, 47, 50mpjaodan 763 1  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   -->wf 5479   ` cfv 5483  (class class class)co 6110   RRcr 9020   -ucneg 9323   NNcn 10031   NN0cn0 10252   ZZcz 10313   Basecbs 13500   Grpcgrp 14716   inv gcminusg 14717  .gcmg 14720   MndHom cmhm 14767    GrpHom cghm 15034
This theorem is referenced by:  ghmcyg  15536  mulgrhm2  16819  dchrabs  21075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-seq 11355  df-0g 13758  df-mnd 14721  df-mhm 14769  df-grp 14843  df-minusg 14844  df-mulg 14846  df-ghm 15035
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