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Theorem ghmmulg 14981
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 14979 . . . . . 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 14885 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
61, 5syl3an1 1217 . . . . 5  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
763expa 1153 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  NN0 )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
87an32s 780 . . 3  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( F `  ( N  .x.  X ) )  =  ( N 
.X.  ( F `  X ) ) )
983adantl2 1114 . 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 960 . . . . . . . 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 10192 . . . . . . . 8  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
1312ad2antll 710 . . . . . . 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 962 . . . . . . 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 14885 . . . . . . 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 1184 . . . . . 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 5699 . . . . 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 14971 . . . . . . . 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 10267 . . . . . . . 8  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
2120ad2antll 710 . . . . . . 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 14870 . . . . . . 7  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
2319, 21, 14, 22syl3anc 1184 . . . . . 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 2412 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
25 eqid 2412 . . . . . . 7  |-  ( inv g `  H )  =  ( inv g `  H )
262, 24, 25ghminv 14976 . . . . . 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 643 . . . . 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 14972 . . . . . . 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 2412 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
312, 30ghmf 14973 . . . . . . . 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 5838 . . . . . 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 14871 . . . . . 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 1184 . . . . 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 2454 . . . 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 14871 . . . . . . 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 1184 . . . . . 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 733 . . . . . . . . 9  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4039recnd 9078 . . . . . . . 8  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4140negnegd 9366 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N
)
4241oveq1d 6063 . . . . . 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 2446 . . . . 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 5699 . . . 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 2446 . . 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 6063 . . 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 2446 . 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 958 . . 3  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
49 elznn0nn 10259 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
5048, 49sylib 189 . 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 762 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 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   -->wf 5417   ` cfv 5421  (class class class)co 6048   RRcr 8953   -ucneg 9256   NNcn 9964   NN0cn0 10185   ZZcz 10246   Basecbs 13432   Grpcgrp 14648   inv gcminusg 14649  .gcmg 14652   MndHom cmhm 14699    GrpHom cghm 14966
This theorem is referenced by:  ghmcyg  15468  mulgrhm2  16751  dchrabs  21005
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-seq 11287  df-0g 13690  df-mnd 14653  df-mhm 14701  df-grp 14775  df-minusg 14776  df-mulg 14778  df-ghm 14967
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