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Theorem mhmmulg 15988
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 6292 . . . . . . 7  |-  ( n  =  0  ->  (
n  .x.  X )  =  ( 0  .x. 
X ) )
21fveq2d 5870 . . . . . 6  |-  ( n  =  0  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
0  .x.  X )
) )
3 oveq1 6292 . . . . . 6  |-  ( n  =  0  ->  (
n  .X.  ( F `  X ) )  =  ( 0  .X.  ( F `  X )
) )
42, 3eqeq12d 2489 . . . . 5  |-  ( n  =  0  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( 0  .x.  X
) )  =  ( 0  .X.  ( F `  X ) ) ) )
54imbi2d 316 . . . 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 6292 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
76fveq2d 5870 . . . . . 6  |-  ( n  =  m  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
m  .x.  X )
) )
8 oveq1 6292 . . . . . 6  |-  ( n  =  m  ->  (
n  .X.  ( F `  X ) )  =  ( m  .X.  ( F `  X )
) )
97, 8eqeq12d 2489 . . . . 5  |-  ( n  =  m  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( m  .x.  X
) )  =  ( m  .X.  ( F `  X ) ) ) )
109imbi2d 316 . . . 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 6292 . . . . . . 7  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  X )  =  ( ( m  +  1 )  .x.  X ) )
1211fveq2d 5870 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
( m  +  1 )  .x.  X ) ) )
13 oveq1 6292 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  (
n  .X.  ( F `  X ) )  =  ( ( m  + 
1 )  .X.  ( F `  X )
) )
1412, 13eqeq12d 2489 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( ( m  +  1 )  .X.  ( F `  X ) ) ) )
1514imbi2d 316 . . . 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 6292 . . . . . . 7  |-  ( n  =  N  ->  (
n  .x.  X )  =  ( N  .x.  X ) )
1716fveq2d 5870 . . . . . 6  |-  ( n  =  N  ->  ( F `  ( n  .x.  X ) )  =  ( F `  ( N  .x.  X ) ) )
18 oveq1 6292 . . . . . 6  |-  ( n  =  N  ->  (
n  .X.  ( F `  X ) )  =  ( N  .X.  ( F `  X )
) )
1917, 18eqeq12d 2489 . . . . 5  |-  ( n  =  N  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( N  .x.  X
) )  =  ( N  .X.  ( F `  X ) ) ) )
2019imbi2d 316 . . . 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 2467 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
22 eqid 2467 . . . . . . 7  |-  ( 0g
`  H )  =  ( 0g `  H
)
2321, 22mhm0 15797 . . . . . 6  |-  ( F  e.  ( G MndHom  H
)  ->  ( F `  ( 0g `  G
) )  =  ( 0g `  H ) )
2423adantr 465 . . . . 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 15961 . . . . . . 7  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2827adantl 466 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2928fveq2d 5870 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( F `  ( 0g `  G ) ) )
30 eqid 2467 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
3125, 30mhmf 15794 . . . . . . 7  |-  ( F  e.  ( G MndHom  H
)  ->  F : B
--> ( Base `  H
) )
3231ffvelrnda 6022 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  H
) )
33 mhmmulg.t . . . . . . 7  |-  .X.  =  (.g
`  H )
3430, 22, 33mulg0 15961 . . . . . 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 2518 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( 0  .X.  ( F `  X )
) )
37 oveq1 6292 . . . . . . 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 15792 . . . . . . . . . . . 12  |-  ( F  e.  ( G MndHom  H
)  ->  G  e.  Mnd )
3938ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  G  e.  Mnd )
40 simpr 461 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  m  e.  NN0 )
41 simplr 754 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  X  e.  B
)
42 eqid 2467 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
4325, 26, 42mulgnn0p1 15967 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
( m  +  1 )  .x.  X )  =  ( ( m 
.x.  X ) ( +g  `  G ) X ) )
4439, 40, 41, 43syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( ( m  +  1 )  .x.  X )  =  ( ( m  .x.  X
) ( +g  `  G
) X ) )
4544fveq2d 5870 . . . . . . . . 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 753 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  F  e.  ( G MndHom  H ) )
4738ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  G  e.  Mnd )
48 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  m  e.  NN0 )
49 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  X  e.  B )
5025, 26mulgnn0cl 15972 . . . . . . . . . . . 12  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
m  .x.  X )  e.  B )
5147, 48, 49, 50syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  ( m  .x.  X
)  e.  B )
5251an32s 802 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( m  .x.  X )  e.  B
)
53 eqid 2467 . . . . . . . . . . 11  |-  ( +g  `  H )  =  ( +g  `  H )
5425, 42, 53mhmlin 15796 . . . . . . . . . 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 1228 . . . . . . . . 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 2508 . . . . . . . 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 15793 . . . . . . . . . 10  |-  ( F  e.  ( G MndHom  H
)  ->  H  e.  Mnd )
5857ad2antrr 725 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  H  e.  Mnd )
5932adantr 465 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( F `  X )  e.  (
Base `  H )
)
6030, 33, 53mulgnn0p1 15967 . . . . . . . . 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 1228 . . . . . . . 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 2489 . . . . . . 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 221 . . . . . 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 435 . . . . 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 26 . . . 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 10958 . . 3  |-  ( N  e.  NN0  ->  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) ) )
67663impib 1194 . 2  |-  ( ( N  e.  NN0  /\  F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
68673com12 1200 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6285   0cc0 9493   1c1 9494    + caddc 9496   NN0cn0 10796   Basecbs 14493   +g cplusg 14558   0gc0g 14698   Mndcmnd 15729  .gcmg 15734   MndHom cmhm 15787
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 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-seq 12077  df-0g 14700  df-mnd 15735  df-mhm 15789  df-mulg 15874
This theorem is referenced by:  pwsmulg  15998  ghmmulg  16093  evls1varpw  18174  evl1expd  18192  cayhamlem4  19196  dchrfi  23355  lgsqrlem1  23441  lgseisenlem4  23452  dchrisum0flblem1  23518
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