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Theorem mhmmulg 16742
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 6303 . . . . . . 7  |-  ( n  =  0  ->  (
n  .x.  X )  =  ( 0  .x. 
X ) )
21fveq2d 5876 . . . . . 6  |-  ( n  =  0  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
0  .x.  X )
) )
3 oveq1 6303 . . . . . 6  |-  ( n  =  0  ->  (
n  .X.  ( F `  X ) )  =  ( 0  .X.  ( F `  X )
) )
42, 3eqeq12d 2442 . . . . 5  |-  ( n  =  0  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( 0  .x.  X
) )  =  ( 0  .X.  ( F `  X ) ) ) )
54imbi2d 317 . . . 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 6303 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
76fveq2d 5876 . . . . . 6  |-  ( n  =  m  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
m  .x.  X )
) )
8 oveq1 6303 . . . . . 6  |-  ( n  =  m  ->  (
n  .X.  ( F `  X ) )  =  ( m  .X.  ( F `  X )
) )
97, 8eqeq12d 2442 . . . . 5  |-  ( n  =  m  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( m  .x.  X
) )  =  ( m  .X.  ( F `  X ) ) ) )
109imbi2d 317 . . . 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 6303 . . . . . . 7  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  X )  =  ( ( m  +  1 )  .x.  X ) )
1211fveq2d 5876 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
( m  +  1 )  .x.  X ) ) )
13 oveq1 6303 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  (
n  .X.  ( F `  X ) )  =  ( ( m  + 
1 )  .X.  ( F `  X )
) )
1412, 13eqeq12d 2442 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( ( m  +  1 )  .X.  ( F `  X ) ) ) )
1514imbi2d 317 . . . 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 6303 . . . . . . 7  |-  ( n  =  N  ->  (
n  .x.  X )  =  ( N  .x.  X ) )
1716fveq2d 5876 . . . . . 6  |-  ( n  =  N  ->  ( F `  ( n  .x.  X ) )  =  ( F `  ( N  .x.  X ) ) )
18 oveq1 6303 . . . . . 6  |-  ( n  =  N  ->  (
n  .X.  ( F `  X ) )  =  ( N  .X.  ( F `  X )
) )
1917, 18eqeq12d 2442 . . . . 5  |-  ( n  =  N  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( N  .x.  X
) )  =  ( N  .X.  ( F `  X ) ) ) )
2019imbi2d 317 . . . 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 2420 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
22 eqid 2420 . . . . . . 7  |-  ( 0g
`  H )  =  ( 0g `  H
)
2321, 22mhm0 16542 . . . . . 6  |-  ( F  e.  ( G MndHom  H
)  ->  ( F `  ( 0g `  G
) )  =  ( 0g `  H ) )
2423adantr 466 . . . . 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 16715 . . . . . . 7  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2827adantl 467 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2928fveq2d 5876 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( F `  ( 0g `  G ) ) )
30 eqid 2420 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
3125, 30mhmf 16539 . . . . . . 7  |-  ( F  e.  ( G MndHom  H
)  ->  F : B
--> ( Base `  H
) )
3231ffvelrnda 6028 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  H
) )
33 mhmmulg.t . . . . . . 7  |-  .X.  =  (.g
`  H )
3430, 22, 33mulg0 16715 . . . . . 6  |-  ( ( F `  X )  e.  ( Base `  H
)  ->  ( 0 
.X.  ( F `  X ) )  =  ( 0g `  H
) )
3532, 34syl 17 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  (
0  .X.  ( F `  X ) )  =  ( 0g `  H
) )
3624, 29, 353eqtr4d 2471 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( 0  .X.  ( F `  X )
) )
37 oveq1 6303 . . . . . . 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 16537 . . . . . . . . . . . 12  |-  ( F  e.  ( G MndHom  H
)  ->  G  e.  Mnd )
3938ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  G  e.  Mnd )
40 simpr 462 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  m  e.  NN0 )
41 simplr 760 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  X  e.  B
)
42 eqid 2420 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
4325, 26, 42mulgnn0p1 16721 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
( m  +  1 )  .x.  X )  =  ( ( m 
.x.  X ) ( +g  `  G ) X ) )
4439, 40, 41, 43syl3anc 1264 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( ( m  +  1 )  .x.  X )  =  ( ( m  .x.  X
) ( +g  `  G
) X ) )
4544fveq2d 5876 . . . . . . . . 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 758 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  F  e.  ( G MndHom  H ) )
4738ad2antrr 730 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  G  e.  Mnd )
48 simplr 760 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  m  e.  NN0 )
49 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  X  e.  B )
5025, 26mulgnn0cl 16726 . . . . . . . . . . . 12  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
m  .x.  X )  e.  B )
5147, 48, 49, 50syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  ( m  .x.  X
)  e.  B )
5251an32s 811 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( m  .x.  X )  e.  B
)
53 eqid 2420 . . . . . . . . . . 11  |-  ( +g  `  H )  =  ( +g  `  H )
5425, 42, 53mhmlin 16541 . . . . . . . . . 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 1264 . . . . . . . . 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 2461 . . . . . . . 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 16538 . . . . . . . . . 10  |-  ( F  e.  ( G MndHom  H
)  ->  H  e.  Mnd )
5857ad2antrr 730 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  H  e.  Mnd )
5932adantr 466 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( F `  X )  e.  (
Base `  H )
)
6030, 33, 53mulgnn0p1 16721 . . . . . . . . 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 1264 . . . . . . . 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 2442 . . . . . . 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 224 . . . . . 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 436 . . . . 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 29 . . . 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 11019 . . 3  |-  ( N  e.  NN0  ->  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) ) )
67663impib 1203 . 2  |-  ( ( N  e.  NN0  /\  F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
68673com12 1209 1  |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   ` cfv 5592  (class class class)co 6296   0cc0 9528   1c1 9529    + caddc 9531   NN0cn0 10858   Basecbs 15081   +g cplusg 15150   0gc0g 15298   Mndcmnd 16487   MndHom cmhm 16532  .gcmg 16624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-seq 12200  df-0g 15300  df-mgm 16440  df-sgrp 16479  df-mnd 16489  df-mhm 16534  df-mulg 16628
This theorem is referenced by:  pwsmulg  16752  ghmmulg  16847  evls1varpw  18856  evl1expd  18874  cayhamlem4  19849  dchrfi  24085  lgsqrlem1  24171  lgseisenlem4  24182  dchrisum0flblem1  24248
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