MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mhmmulg Unicode version

Theorem mhmmulg 14881
Description: A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
mhmmulg.b  |-  B  =  ( Base `  G
)
mhmmulg.s  |-  .x.  =  (.g
`  G )
mhmmulg.t  |-  .X.  =  (.g
`  H )
Assertion
Ref Expression
mhmmulg  |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )

Proof of Theorem mhmmulg
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6051 . . . . . . 7  |-  ( n  =  0  ->  (
n  .x.  X )  =  ( 0  .x. 
X ) )
21fveq2d 5695 . . . . . 6  |-  ( n  =  0  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
0  .x.  X )
) )
3 oveq1 6051 . . . . . 6  |-  ( n  =  0  ->  (
n  .X.  ( F `  X ) )  =  ( 0  .X.  ( F `  X )
) )
42, 3eqeq12d 2422 . . . . 5  |-  ( n  =  0  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( 0  .x.  X
) )  =  ( 0  .X.  ( F `  X ) ) ) )
54imbi2d 308 . . . 4  |-  ( n  =  0  ->  (
( ( F  e.  ( G MndHom  H )  /\  X  e.  B
)  ->  ( F `  ( n  .x.  X
) )  =  ( n  .X.  ( F `  X ) ) )  <-> 
( ( F  e.  ( G MndHom  H )  /\  X  e.  B
)  ->  ( F `  ( 0  .x.  X
) )  =  ( 0  .X.  ( F `  X ) ) ) ) )
6 oveq1 6051 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
76fveq2d 5695 . . . . . 6  |-  ( n  =  m  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
m  .x.  X )
) )
8 oveq1 6051 . . . . . 6  |-  ( n  =  m  ->  (
n  .X.  ( F `  X ) )  =  ( m  .X.  ( F `  X )
) )
97, 8eqeq12d 2422 . . . . 5  |-  ( n  =  m  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( m  .x.  X
) )  =  ( m  .X.  ( F `  X ) ) ) )
109imbi2d 308 . . . 4  |-  ( n  =  m  ->  (
( ( F  e.  ( G MndHom  H )  /\  X  e.  B
)  ->  ( F `  ( n  .x.  X
) )  =  ( n  .X.  ( F `  X ) ) )  <-> 
( ( F  e.  ( G MndHom  H )  /\  X  e.  B
)  ->  ( F `  ( m  .x.  X
) )  =  ( m  .X.  ( F `  X ) ) ) ) )
11 oveq1 6051 . . . . . . 7  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  X )  =  ( ( m  +  1 )  .x.  X ) )
1211fveq2d 5695 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
( m  +  1 )  .x.  X ) ) )
13 oveq1 6051 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  (
n  .X.  ( F `  X ) )  =  ( ( m  + 
1 )  .X.  ( F `  X )
) )
1412, 13eqeq12d 2422 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( ( m  +  1 )  .X.  ( F `  X ) ) ) )
1514imbi2d 308 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  (
( ( F  e.  ( G MndHom  H )  /\  X  e.  B
)  ->  ( F `  ( n  .x.  X
) )  =  ( n  .X.  ( F `  X ) ) )  <-> 
( ( F  e.  ( G MndHom  H )  /\  X  e.  B
)  ->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( ( m  +  1 )  .X.  ( F `  X ) ) ) ) )
16 oveq1 6051 . . . . . . 7  |-  ( n  =  N  ->  (
n  .x.  X )  =  ( N  .x.  X ) )
1716fveq2d 5695 . . . . . 6  |-  ( n  =  N  ->  ( F `  ( n  .x.  X ) )  =  ( F `  ( N  .x.  X ) ) )
18 oveq1 6051 . . . . . 6  |-  ( n  =  N  ->  (
n  .X.  ( F `  X ) )  =  ( N  .X.  ( F `  X )
) )
1917, 18eqeq12d 2422 . . . . 5  |-  ( n  =  N  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( N  .x.  X
) )  =  ( N  .X.  ( F `  X ) ) ) )
2019imbi2d 308 . . . 4  |-  ( n  =  N  ->  (
( ( F  e.  ( G MndHom  H )  /\  X  e.  B
)  ->  ( F `  ( n  .x.  X
) )  =  ( n  .X.  ( F `  X ) ) )  <-> 
( ( F  e.  ( G MndHom  H )  /\  X  e.  B
)  ->  ( F `  ( N  .x.  X
) )  =  ( N  .X.  ( F `  X ) ) ) ) )
21 eqid 2408 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
22 eqid 2408 . . . . . . 7  |-  ( 0g
`  H )  =  ( 0g `  H
)
2321, 22mhm0 14705 . . . . . 6  |-  ( F  e.  ( G MndHom  H
)  ->  ( F `  ( 0g `  G
) )  =  ( 0g `  H ) )
2423adantr 452 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0g `  G ) )  =  ( 0g `  H
) )
25 mhmmulg.b . . . . . . . 8  |-  B  =  ( Base `  G
)
26 mhmmulg.s . . . . . . . 8  |-  .x.  =  (.g
`  G )
2725, 21, 26mulg0 14854 . . . . . . 7  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2827adantl 453 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2928fveq2d 5695 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( F `  ( 0g `  G ) ) )
30 eqid 2408 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
3125, 30mhmf 14702 . . . . . . 7  |-  ( F  e.  ( G MndHom  H
)  ->  F : B
--> ( Base `  H
) )
3231ffvelrnda 5833 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  H
) )
33 mhmmulg.t . . . . . . 7  |-  .X.  =  (.g
`  H )
3430, 22, 33mulg0 14854 . . . . . 6  |-  ( ( F `  X )  e.  ( Base `  H
)  ->  ( 0 
.X.  ( F `  X ) )  =  ( 0g `  H
) )
3532, 34syl 16 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  (
0  .X.  ( F `  X ) )  =  ( 0g `  H
) )
3624, 29, 353eqtr4d 2450 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( 0  .X.  ( F `  X )
) )
37 oveq1 6051 . . . . . . 7  |-  ( ( F `  ( m 
.x.  X ) )  =  ( m  .X.  ( F `  X ) )  ->  ( ( F `  ( m  .x.  X ) ) ( +g  `  H ) ( F `  X
) )  =  ( ( m  .X.  ( F `  X )
) ( +g  `  H
) ( F `  X ) ) )
38 mhmrcl1 14700 . . . . . . . . . . . 12  |-  ( F  e.  ( G MndHom  H
)  ->  G  e.  Mnd )
3938ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  G  e.  Mnd )
40 simpr 448 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  m  e.  NN0 )
41 simplr 732 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  X  e.  B
)
42 eqid 2408 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
4325, 26, 42mulgnn0p1 14860 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
( m  +  1 )  .x.  X )  =  ( ( m 
.x.  X ) ( +g  `  G ) X ) )
4439, 40, 41, 43syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( ( m  +  1 )  .x.  X )  =  ( ( m  .x.  X
) ( +g  `  G
) X ) )
4544fveq2d 5695 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( F `  ( ( m  .x.  X ) ( +g  `  G
) X ) ) )
46 simpll 731 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  F  e.  ( G MndHom  H ) )
4738ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  G  e.  Mnd )
48 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  m  e.  NN0 )
49 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  X  e.  B )
5025, 26mulgnn0cl 14865 . . . . . . . . . . . 12  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
m  .x.  X )  e.  B )
5147, 48, 49, 50syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  ( m  .x.  X
)  e.  B )
5251an32s 780 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( m  .x.  X )  e.  B
)
53 eqid 2408 . . . . . . . . . . 11  |-  ( +g  `  H )  =  ( +g  `  H )
5425, 42, 53mhmlin 14704 . . . . . . . . . 10  |-  ( ( F  e.  ( G MndHom  H )  /\  (
m  .x.  X )  e.  B  /\  X  e.  B )  ->  ( F `  ( (
m  .x.  X )
( +g  `  G ) X ) )  =  ( ( F `  ( m  .x.  X ) ) ( +g  `  H
) ( F `  X ) ) )
5546, 52, 41, 54syl3anc 1184 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( F `  ( ( m  .x.  X ) ( +g  `  G ) X ) )  =  ( ( F `  ( m 
.x.  X ) ) ( +g  `  H
) ( F `  X ) ) )
5645, 55eqtrd 2440 . . . . . . . 8  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( ( F `  (
m  .x.  X )
) ( +g  `  H
) ( F `  X ) ) )
57 mhmrcl2 14701 . . . . . . . . . 10  |-  ( F  e.  ( G MndHom  H
)  ->  H  e.  Mnd )
5857ad2antrr 707 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  H  e.  Mnd )
5932adantr 452 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( F `  X )  e.  (
Base `  H )
)
6030, 33, 53mulgnn0p1 14860 . . . . . . . . 9  |-  ( ( H  e.  Mnd  /\  m  e.  NN0  /\  ( F `  X )  e.  ( Base `  H
) )  ->  (
( m  +  1 )  .X.  ( F `  X ) )  =  ( ( m  .X.  ( F `  X ) ) ( +g  `  H
) ( F `  X ) ) )
6158, 40, 59, 60syl3anc 1184 . . . . . . . 8  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( ( m  +  1 )  .X.  ( F `  X ) )  =  ( ( m  .X.  ( F `  X ) ) ( +g  `  H ) ( F `  X
) ) )
6256, 61eqeq12d 2422 . . . . . . 7  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( ( F `
 ( ( m  +  1 )  .x.  X ) )  =  ( ( m  + 
1 )  .X.  ( F `  X )
)  <->  ( ( F `
 ( m  .x.  X ) ) ( +g  `  H ) ( F `  X
) )  =  ( ( m  .X.  ( F `  X )
) ( +g  `  H
) ( F `  X ) ) ) )
6337, 62syl5ibr 213 . . . . . 6  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( ( F `
 ( m  .x.  X ) )  =  ( m  .X.  ( F `  X )
)  ->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( ( m  +  1 )  .X.  ( F `  X ) ) ) )
6463expcom 425 . . . . 5  |-  ( m  e.  NN0  ->  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  (
( F `  (
m  .x.  X )
)  =  ( m 
.X.  ( F `  X ) )  -> 
( F `  (
( m  +  1 )  .x.  X ) )  =  ( ( m  +  1 ) 
.X.  ( F `  X ) ) ) ) )
6564a2d 24 . . . 4  |-  ( m  e.  NN0  ->  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  (
m  .x.  X )
)  =  ( m 
.X.  ( F `  X ) ) )  ->  ( ( F  e.  ( G MndHom  H
)  /\  X  e.  B )  ->  ( F `  ( (
m  +  1 ) 
.x.  X ) )  =  ( ( m  +  1 )  .X.  ( F `  X ) ) ) ) )
665, 10, 15, 20, 36, 65nn0ind 10326 . . 3  |-  ( N  e.  NN0  ->  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) ) )
67663impib 1151 . 2  |-  ( ( N  e.  NN0  /\  F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
68673com12 1157 1  |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   0cc0 8950   1c1 8951    + caddc 8953   NN0cn0 10181   Basecbs 13428   +g cplusg 13488   0gc0g 13682   Mndcmnd 14643  .gcmg 14648   MndHom cmhm 14695
This theorem is referenced by:  pwsmulg  14891  ghmmulg  14977  evl1expd  19915  dchrfi  20996  lgsqrlem1  21082  lgseisenlem4  21093  dchrisum0flblem1  21159
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-seq 11283  df-0g 13686  df-mnd 14649  df-mhm 14697  df-mulg 14774
  Copyright terms: Public domain W3C validator