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Theorem mhmmulg 14692
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 5949 . . . . . . 7  |-  ( n  =  0  ->  (
n  .x.  X )  =  ( 0  .x. 
X ) )
21fveq2d 5609 . . . . . 6  |-  ( n  =  0  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
0  .x.  X )
) )
3 oveq1 5949 . . . . . 6  |-  ( n  =  0  ->  (
n  .X.  ( F `  X ) )  =  ( 0  .X.  ( F `  X )
) )
42, 3eqeq12d 2372 . . . . 5  |-  ( n  =  0  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( 0  .x.  X
) )  =  ( 0  .X.  ( F `  X ) ) ) )
54imbi2d 307 . . . 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 5949 . . . . . . 7  |-  ( n  =  m  ->  (
n  .x.  X )  =  ( m  .x.  X ) )
76fveq2d 5609 . . . . . 6  |-  ( n  =  m  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
m  .x.  X )
) )
8 oveq1 5949 . . . . . 6  |-  ( n  =  m  ->  (
n  .X.  ( F `  X ) )  =  ( m  .X.  ( F `  X )
) )
97, 8eqeq12d 2372 . . . . 5  |-  ( n  =  m  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( m  .x.  X
) )  =  ( m  .X.  ( F `  X ) ) ) )
109imbi2d 307 . . . 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 5949 . . . . . . 7  |-  ( n  =  ( m  + 
1 )  ->  (
n  .x.  X )  =  ( ( m  +  1 )  .x.  X ) )
1211fveq2d 5609 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  ( F `  ( n  .x.  X ) )  =  ( F `  (
( m  +  1 )  .x.  X ) ) )
13 oveq1 5949 . . . . . 6  |-  ( n  =  ( m  + 
1 )  ->  (
n  .X.  ( F `  X ) )  =  ( ( m  + 
1 )  .X.  ( F `  X )
) )
1412, 13eqeq12d 2372 . . . . 5  |-  ( n  =  ( m  + 
1 )  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( ( m  +  1 )  .X.  ( F `  X ) ) ) )
1514imbi2d 307 . . . 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 5949 . . . . . . 7  |-  ( n  =  N  ->  (
n  .x.  X )  =  ( N  .x.  X ) )
1716fveq2d 5609 . . . . . 6  |-  ( n  =  N  ->  ( F `  ( n  .x.  X ) )  =  ( F `  ( N  .x.  X ) ) )
18 oveq1 5949 . . . . . 6  |-  ( n  =  N  ->  (
n  .X.  ( F `  X ) )  =  ( N  .X.  ( F `  X )
) )
1917, 18eqeq12d 2372 . . . . 5  |-  ( n  =  N  ->  (
( F `  (
n  .x.  X )
)  =  ( n 
.X.  ( F `  X ) )  <->  ( F `  ( N  .x.  X
) )  =  ( N  .X.  ( F `  X ) ) ) )
2019imbi2d 307 . . . 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 2358 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
22 eqid 2358 . . . . . . 7  |-  ( 0g
`  H )  =  ( 0g `  H
)
2321, 22mhm0 14516 . . . . . 6  |-  ( F  e.  ( G MndHom  H
)  ->  ( F `  ( 0g `  G
) )  =  ( 0g `  H ) )
2423adantr 451 . . . . 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 14665 . . . . . . 7  |-  ( X  e.  B  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2827adantl 452 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  (
0  .x.  X )  =  ( 0g `  G ) )
2928fveq2d 5609 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( F `  ( 0g `  G ) ) )
30 eqid 2358 . . . . . . . 8  |-  ( Base `  H )  =  (
Base `  H )
3125, 30mhmf 14513 . . . . . . 7  |-  ( F  e.  ( G MndHom  H
)  ->  F : B
--> ( Base `  H
) )
32 ffvelrn 5743 . . . . . . 7  |-  ( ( F : B --> ( Base `  H )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  H
) )
3331, 32sylan 457 . . . . . 6  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  H
) )
34 mhmmulg.t . . . . . . 7  |-  .X.  =  (.g
`  H )
3530, 22, 34mulg0 14665 . . . . . 6  |-  ( ( F `  X )  e.  ( Base `  H
)  ->  ( 0 
.X.  ( F `  X ) )  =  ( 0g `  H
) )
3633, 35syl 15 . . . . 5  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  (
0  .X.  ( F `  X ) )  =  ( 0g `  H
) )
3724, 29, 363eqtr4d 2400 . . . 4  |-  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( 0  .x.  X ) )  =  ( 0  .X.  ( F `  X )
) )
38 oveq1 5949 . . . . . . 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 ) ) )
39 mhmrcl1 14511 . . . . . . . . . . . 12  |-  ( F  e.  ( G MndHom  H
)  ->  G  e.  Mnd )
4039ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  G  e.  Mnd )
41 simpr 447 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  m  e.  NN0 )
42 simplr 731 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  X  e.  B
)
43 eqid 2358 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
4425, 26, 43mulgnn0p1 14671 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
( m  +  1 )  .x.  X )  =  ( ( m 
.x.  X ) ( +g  `  G ) X ) )
4540, 41, 42, 44syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( ( m  +  1 )  .x.  X )  =  ( ( m  .x.  X
) ( +g  `  G
) X ) )
4645fveq2d 5609 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( F `  ( ( m  + 
1 )  .x.  X
) )  =  ( F `  ( ( m  .x.  X ) ( +g  `  G
) X ) ) )
47 simpll 730 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  F  e.  ( G MndHom  H ) )
4839ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  G  e.  Mnd )
49 simplr 731 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  m  e.  NN0 )
50 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  X  e.  B )
5125, 26mulgnn0cl 14676 . . . . . . . . . . . 12  |-  ( ( G  e.  Mnd  /\  m  e.  NN0  /\  X  e.  B )  ->  (
m  .x.  X )  e.  B )
5248, 49, 50, 51syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( G MndHom  H )  /\  m  e.  NN0 )  /\  X  e.  B )  ->  ( m  .x.  X
)  e.  B )
5352an32s 779 . . . . . . . . . 10  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( m  .x.  X )  e.  B
)
54 eqid 2358 . . . . . . . . . . 11  |-  ( +g  `  H )  =  ( +g  `  H )
5525, 43, 54mhmlin 14515 . . . . . . . . . 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 ) ) )
5647, 53, 42, 55syl3anc 1182 . . . . . . . . 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 ) ) )
5746, 56eqtrd 2390 . . . . . . . 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 ) ) )
58 mhmrcl2 14512 . . . . . . . . . 10  |-  ( F  e.  ( G MndHom  H
)  ->  H  e.  Mnd )
5958ad2antrr 706 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  H  e.  Mnd )
6033adantr 451 . . . . . . . . 9  |-  ( ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  /\  m  e.  NN0 )  ->  ( F `  X )  e.  (
Base `  H )
)
6130, 34, 54mulgnn0p1 14671 . . . . . . . . 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 ) ) )
6259, 41, 60, 61syl3anc 1182 . . . . . . . 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
) ) )
6357, 62eqeq12d 2372 . . . . . . 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 ) ) ) )
6438, 63syl5ibr 212 . . . . . 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 ) ) ) )
6564expcom 424 . . . . 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 ) ) ) ) )
6665a2d 23 . . . 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 ) ) ) ) )
675, 10, 15, 20, 37, 66nn0ind 10197 . . 3  |-  ( N  e.  NN0  ->  ( ( F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) ) )
68673impib 1149 . 2  |-  ( ( N  e.  NN0  /\  F  e.  ( G MndHom  H )  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X )
) )
69683com12 1155 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710   -->wf 5330   ` cfv 5334  (class class class)co 5942   0cc0 8824   1c1 8825    + caddc 8827   NN0cn0 10054   Basecbs 13239   +g cplusg 13299   0gc0g 13493   Mndcmnd 14454  .gcmg 14459   MndHom cmhm 14506
This theorem is referenced by:  pwsmulg  14702  ghmmulg  14788  evl1expd  19519  dchrfi  20600  lgsqrlem1  20686  lgseisenlem4  20697  dchrisum0flblem1  20763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-seq 11136  df-0g 13497  df-mnd 14460  df-mhm 14508  df-mulg 14585
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