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Theorem ghmmulg 16150
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 16148 . . . . . 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 16045 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
61, 5syl3an1 1260 . . . . 5  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
763expa 1195 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  NN0 )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
87an32s 802 . . 3  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  X  e.  B )  /\  N  e.  NN0 )  ->  ( F `  ( N  .x.  X ) )  =  ( N 
.X.  ( F `  X ) ) )
983adantl2 1152 . 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 998 . . . . . . . 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 10805 . . . . . . . 8  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
1312ad2antll 728 . . . . . . 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 1000 . . . . . . 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 16045 . . . . . . 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 1227 . . . . . 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 5857 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( invg `  H ) `  ( F `  ( -u N  .x.  X ) ) )  =  ( ( invg `  H ) `
 ( -u N  .X.  ( F `  X
) ) ) )
18 ghmgrp1 16140 . . . . . . . 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 10889 . . . . . . . 8  |-  ( -u N  e.  NN  ->  -u N  e.  ZZ )
2120ad2antll 728 . . . . . . 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 16030 . . . . . . 7  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
2319, 21, 14, 22syl3anc 1227 . . . . . 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 2441 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
25 eqid 2441 . . . . . . 7  |-  ( invg `  H )  =  ( invg `  H )
262, 24, 25ghminv 16145 . . . . . 6  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  ( -u N  .x.  X )  e.  B )  -> 
( F `  (
( invg `  G ) `  ( -u N  .x.  X ) ) )  =  ( ( invg `  H ) `  ( F `  ( -u N  .x.  X ) ) ) )
2710, 23, 26syl2anc 661 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  (
( invg `  G ) `  ( -u N  .x.  X ) ) )  =  ( ( invg `  H ) `  ( F `  ( -u N  .x.  X ) ) ) )
28 ghmgrp2 16141 . . . . . . 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 2441 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
312, 30ghmf 16142 . . . . . . . 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 6014 . . . . . 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 16031 . . . . . 6  |-  ( ( H  e.  Grp  /\  -u N  e.  ZZ  /\  ( F `  X )  e.  ( Base `  H
) )  ->  ( -u -u N  .X.  ( F `
 X ) )  =  ( ( invg `  H ) `
 ( -u N  .X.  ( F `  X
) ) ) )
3529, 21, 33, 34syl3anc 1227 . . . . 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 )
)  =  ( ( invg `  H
) `  ( -u N  .X.  ( F `  X
) ) ) )
3617, 27, 353eqtr4d 2492 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  (
( invg `  G ) `  ( -u N  .x.  X ) ) )  =  (
-u -u N  .X.  ( F `  X )
) )
372, 3, 24mulgneg 16031 . . . . . . 7  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u -u N  .x.  X )  =  ( ( invg `  G ) `  ( -u N  .x.  X ) ) )
3819, 21, 14, 37syl3anc 1227 . . . . . 6  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( -u -u N  .x.  X
)  =  ( ( invg `  G
) `  ( -u N  .x.  X ) ) )
39 simprl 755 . . . . . . . . 9  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  RR )
4039recnd 9622 . . . . . . . 8  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  N  e.  CC )
4140negnegd 9924 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N
)
4241oveq1d 6293 . . . . . 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 2484 . . . . 5  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( ( invg `  G ) `  ( -u N  .x.  X ) )  =  ( N 
.x.  X ) )
4443fveq2d 5857 . . . 4  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  -> 
( F `  (
( invg `  G ) `  ( -u N  .x.  X ) ) )  =  ( F `  ( N 
.x.  X ) ) )
4536, 44eqtr3d 2484 . . 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 6293 . . 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 2484 . 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 996 . . 3  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
49 elznn0nn 10881 . . 3  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
5048, 49sylib 196 . 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 784 1  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   -->wf 5571   ` cfv 5575  (class class class)co 6278   RRcr 9491   -ucneg 9808   NNcn 10539   NN0cn0 10798   ZZcz 10867   Basecbs 14506   MndHom cmhm 15835   Grpcgrp 15924   invgcminusg 15925  .gcmg 15927    GrpHom cghm 16135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-er 7310  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-n0 10799  df-z 10868  df-uz 11088  df-fz 11679  df-seq 12084  df-0g 14713  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-mhm 15837  df-grp 15928  df-minusg 15929  df-mulg 15931  df-ghm 16136
This theorem is referenced by:  ghmcyg  16769  mulgrhm2  18403  mulgrhm2OLD  18406  dchrabs  23404
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