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Theorem ghmmulg 16084
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 16082 . . . . . 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 15984 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
61, 5syl3an1 1261 . . . . 5  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
763expa 1196 . . . 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 1153 . 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 999 . . . . . . . 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 10802 . . . . . . . 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 1001 . . . . . . 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 15984 . . . . . . 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 1228 . . . . . 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 5870 . . . . 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 16074 . . . . . . . 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 10886 . . . . . . . 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 15969 . . . . . . 7  |-  ( ( G  e.  Grp  /\  -u N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  e.  B
)
2319, 21, 14, 22syl3anc 1228 . . . . . 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 2467 . . . . . . 7  |-  ( invg `  G )  =  ( invg `  G )
25 eqid 2467 . . . . . . 7  |-  ( invg `  H )  =  ( invg `  H )
262, 24, 25ghminv 16079 . . . . . 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 16075 . . . . . . 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 2467 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
312, 30ghmf 16076 . . . . . . . 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 6022 . . . . . 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 15970 . . . . . 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 1228 . . . . 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 2518 . . . 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 15970 . . . . . . 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 1228 . . . . . 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 9921 . . . . . . 7  |-  ( ( ( F  e.  ( G  GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( N  e.  RR  /\  -u N  e.  NN ) )  ->  -u -u N  =  N
)
4241oveq1d 6299 . . . . . 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 2510 . . . . 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 5870 . . . 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 2510 . . 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 6299 . . 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 2510 . 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 997 . . 3  |-  ( ( F  e.  ( G 
GrpHom  H )  /\  N  e.  ZZ  /\  X  e.  B )  ->  N  e.  ZZ )
49 elznn0nn 10878 . . 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 973    = wceq 1379    e. wcel 1767   -->wf 5584   ` cfv 5588  (class class class)co 6284   RRcr 9491   -ucneg 9806   NNcn 10536   NN0cn0 10795   ZZcz 10864   Basecbs 14490   Grpcgrp 15727   invgcminusg 15728  .gcmg 15731   MndHom cmhm 15784    GrpHom cghm 16069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  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 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-seq 12076  df-0g 14697  df-mnd 15732  df-mhm 15786  df-grp 15867  df-minusg 15868  df-mulg 15870  df-ghm 16070
This theorem is referenced by:  ghmcyg  16701  mulgrhm2  18328  mulgrhm2OLD  18331  dchrabs  23291
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