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Theorem ghmmulg 14711
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 14709 . . . . . 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 14615 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
61, 5syl3an1 1215 . . . . 5  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
763expa 1151 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  NN0 )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
87an32s 779 . . 3  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( F `  ( N  .x.  X ) )  =  ( N 
.X.  ( F `  X ) ) )
983adantl2 1112 . 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 958 . . . . . . . 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 15 . . . . . . 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 9988 . . . . . . . 8  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
1312ad2antll 709 . . . . . . 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 960 . . . . . . 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 14615 . . . . . . 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 1182 . . . . . 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 5545 . . . . 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 14701 . . . . . . . 8  |-  ( F  e.  ( G  GrpHom  H )  ->  G  e.  Grp )
1910, 18syl 15 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  G  e.  Grp )
20 nnz 10061 . . . . . . . 8  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
2120ad2antll 709 . . . . . . 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 14600 . . . . . . 7  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
2319, 21, 14, 22syl3anc 1182 . . . . . 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 2296 . . . . . . 7  |-  ( inv g `  G )  =  ( inv g `  G )
25 eqid 2296 . . . . . . 7  |-  ( inv g `  H )  =  ( inv g `  H )
262, 24, 25ghminv 14706 . . . . . 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 642 . . . . 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 14702 . . . . . . 7  |-  ( F  e.  ( G  GrpHom  H )  ->  H  e.  Grp )
2910, 28syl 15 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  H  e.  Grp )
30 eqid 2296 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
312, 30ghmf 14703 . . . . . . . 8  |-  ( F  e.  ( G  GrpHom  H )  ->  F : B
--> ( Base `  H
) )
3210, 31syl 15 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  F : B --> ( Base `  H ) )
33 ffvelrn 5679 . . . . . . 7  |-  ( ( F : B --> ( Base `  H )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  H
) )
3432, 14, 33syl2anc 642 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  X
)  e.  ( Base `  H ) )
3530, 4, 25mulgneg 14601 . . . . . 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
) ) ) )
3629, 21, 34, 35syl3anc 1182 . . . . 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
) ) ) )
3717, 27, 363eqtr4d 2338 . . . 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 )
) )
382, 3, 24mulgneg 14601 . . . . . . 7  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( ( inv g `  G ) `  ( -u N  .x.  X ) ) )
3919, 21, 14, 38syl3anc 1182 . . . . . 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 ) ) )
40 simprl 732 . . . . . . . . 9  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4140recnd 8877 . . . . . . . 8  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4241negnegd 9164 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N
)
4342oveq1d 5889 . . . . . 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 ) )
4439, 43eqtr3d 2330 . . . . 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 ) )
4544fveq2d 5545 . . . 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 ) ) )
4637, 45eqtr3d 2330 . . 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 ) ) )
4742oveq1d 5889 . . 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 ) ) )
4846, 47eqtr3d 2330 . 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 ) ) )
49 simp2 956 . . 3  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
50 elznn0nn 10053 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
5149, 50sylib 188 . 2  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
529, 48, 51mpjaodan 761 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 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   -ucneg 9054   NNcn 9762   NN0cn0 9981   ZZcz 10040   Basecbs 13164   Grpcgrp 14378   inv gcminusg 14379  .gcmg 14382   MndHom cmhm 14429    GrpHom cghm 14696
This theorem is referenced by:  ghmcyg  15198  mulgrhm2  16477  dchrabs  20515
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-map 6790  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-mhm 14431  df-grp 14505  df-minusg 14506  df-mulg 14508  df-ghm 14697
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