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Theorem mhmmulg 15779
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 6208 . . . . . . 7  |-  ( n  =  0  ->  (
n  .x.  X )  =  ( 0  .x. 
X ) )
21fveq2d 5804 . . . . . 6  |-  ( n  =  0  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
0  .x.  X )
) )
3 oveq1 6208 . . . . . 6  |-  ( n  =  0  ->  (
n  .X.  ( F `  X ) )  =  ( 0  .X.  ( F `  X )
) )
42, 3eqeq12d 2476 . . . . 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 6208 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
76fveq2d 5804 . . . . . 6  |-  ( n  =  m  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
m  .x.  X )
) )
8 oveq1 6208 . . . . . 6  |-  ( n  =  m  ->  (
n  .X.  ( F `  X ) )  =  ( m  .X.  ( F `  X )
) )
97, 8eqeq12d 2476 . . . . 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 6208 . . . . . . 7  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  X )  =  ( ( m  +  1 )  .x.  X ) )
1211fveq2d 5804 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
( m  +  1 )  .x.  X ) ) )
13 oveq1 6208 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  (
n  .X.  ( F `  X ) )  =  ( ( m  + 
1 )  .X.  ( F `  X )
) )
1412, 13eqeq12d 2476 . . . . 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 6208 . . . . . . 7  |-  ( n  =  N  ->  (
n  .x.  X )  =  ( N  .x.  X ) )
1716fveq2d 5804 . . . . . 6  |-  ( n  =  N  ->  ( F `  ( n  .x.  X ) )  =  ( F `  ( N  .x.  X ) ) )
18 oveq1 6208 . . . . . 6  |-  ( n  =  N  ->  (
n  .X.  ( F `  X ) )  =  ( N  .X.  ( F `  X )
) )
1917, 18eqeq12d 2476 . . . . 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 2454 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
22 eqid 2454 . . . . . . 7  |-  ( 0g
`  H )  =  ( 0g `  H
)
2321, 22mhm0 15592 . . . . . 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 15752 . . . . . . 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 5804 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( F `  ( 0g `  G ) ) )
30 eqid 2454 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
3125, 30mhmf 15589 . . . . . . 7  |-  ( F  e.  ( G MndHom  H
)  ->  F : B
--> ( Base `  H
) )
3231ffvelrnda 5953 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  H
) )
33 mhmmulg.t . . . . . . 7  |-  .X.  =  (.g
`  H )
3430, 22, 33mulg0 15752 . . . . . 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 2505 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( 0  .X.  ( F `  X )
) )
37 oveq1 6208 . . . . . . 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 15587 . . . . . . . . . . . 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 2454 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
4325, 26, 42mulgnn0p1 15758 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
( m  +  1 )  .x.  X )  =  ( ( m 
.x.  X ) ( +g  `  G ) X ) )
4439, 40, 41, 43syl3anc 1219 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( ( m  +  1 )  .x.  X )  =  ( ( m  .x.  X
) ( +g  `  G
) X ) )
4544fveq2d 5804 . . . . . . . . 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 15763 . . . . . . . . . . . 12  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
m  .x.  X )  e.  B )
5147, 48, 49, 50syl3anc 1219 . . . . . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( +g  `  H )  =  ( +g  `  H )
5425, 42, 53mhmlin 15591 . . . . . . . . . 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 1219 . . . . . . . . 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 2495 . . . . . . . 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 15588 . . . . . . . . . 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 15758 . . . . . . . . 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 1219 . . . . . . . 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 2476 . . . . . . 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 10850 . . 3  |-  ( N  e.  NN0  ->  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) ) )
67663impib 1186 . 2  |-  ( ( N  e.  NN0  /\  F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
68673com12 1192 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 965    = wceq 1370    e. wcel 1758   ` cfv 5527  (class class class)co 6201   0cc0 9394   1c1 9395    + caddc 9397   NN0cn0 10691   Basecbs 14293   +g cplusg 14358   0gc0g 14498   Mndcmnd 15529  .gcmg 15534   MndHom cmhm 15582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-seq 11925  df-0g 14500  df-mnd 15535  df-mhm 15584  df-mulg 15668
This theorem is referenced by:  pwsmulg  15789  ghmmulg  15879  evls1varpw  17887  evl1expd  17905  dchrfi  22728  lgsqrlem1  22814  lgseisenlem4  22825  dchrisum0flblem1  22891  cayhamlem4  31376
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