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Theorem mulgghm 15406
Description: The map from  x to  n x for a fixed integer  n is a group homomorphism if the group is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
mulgmhm.b  |-  B  =  ( Base `  G
)
mulgmhm.m  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
mulgghm  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  (
x  e.  B  |->  ( M  .x.  x ) )  e.  ( G 
GrpHom  G ) )
Distinct variable groups:    x, B    x, G    x, M    x,  .x.

Proof of Theorem mulgghm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulgmhm.b . 2  |-  B  =  ( Base `  G
)
2 eqid 2404 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
3 ablgrp 15372 . . 3  |-  ( G  e.  Abel  ->  G  e. 
Grp )
43adantr 452 . 2  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  G  e.  Grp )
5 mulgmhm.m . . . . . 6  |-  .x.  =  (.g
`  G )
61, 5mulgcl 14862 . . . . 5  |-  ( ( G  e.  Grp  /\  M  e.  ZZ  /\  x  e.  B )  ->  ( M  .x.  x )  e.  B )
73, 6syl3an1 1217 . . . 4  |-  ( ( G  e.  Abel  /\  M  e.  ZZ  /\  x  e.  B )  ->  ( M  .x.  x )  e.  B )
873expa 1153 . . 3  |-  ( ( ( G  e.  Abel  /\  M  e.  ZZ )  /\  x  e.  B
)  ->  ( M  .x.  x )  e.  B
)
9 eqid 2404 . . 3  |-  ( x  e.  B  |->  ( M 
.x.  x ) )  =  ( x  e.  B  |->  ( M  .x.  x ) )
108, 9fmptd 5852 . 2  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  (
x  e.  B  |->  ( M  .x.  x ) ) : B --> B )
11 3anass 940 . . . . 5  |-  ( ( M  e.  ZZ  /\  y  e.  B  /\  z  e.  B )  <->  ( M  e.  ZZ  /\  ( y  e.  B  /\  z  e.  B
) ) )
121, 5, 2mulgdi 15404 . . . . 5  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  y  e.  B  /\  z  e.  B ) )  -> 
( M  .x.  (
y ( +g  `  G
) z ) )  =  ( ( M 
.x.  y ) ( +g  `  G ) ( M  .x.  z
) ) )
1311, 12sylan2br 463 . . . 4  |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  (
y  e.  B  /\  z  e.  B )
) )  ->  ( M  .x.  ( y ( +g  `  G ) z ) )  =  ( ( M  .x.  y ) ( +g  `  G ) ( M 
.x.  z ) ) )
1413anassrs 630 . . 3  |-  ( ( ( G  e.  Abel  /\  M  e.  ZZ )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( M  .x.  (
y ( +g  `  G
) z ) )  =  ( ( M 
.x.  y ) ( +g  `  G ) ( M  .x.  z
) ) )
151, 2grpcl 14773 . . . . . 6  |-  ( ( G  e.  Grp  /\  y  e.  B  /\  z  e.  B )  ->  ( y ( +g  `  G ) z )  e.  B )
16153expb 1154 . . . . 5  |-  ( ( G  e.  Grp  /\  ( y  e.  B  /\  z  e.  B
) )  ->  (
y ( +g  `  G
) z )  e.  B )
174, 16sylan 458 . . . 4  |-  ( ( ( G  e.  Abel  /\  M  e.  ZZ )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( y ( +g  `  G ) z )  e.  B )
18 oveq2 6048 . . . . 5  |-  ( x  =  ( y ( +g  `  G ) z )  ->  ( M  .x.  x )  =  ( M  .x.  (
y ( +g  `  G
) z ) ) )
19 ovex 6065 . . . . 5  |-  ( M 
.x.  ( y ( +g  `  G ) z ) )  e. 
_V
2018, 9, 19fvmpt 5765 . . . 4  |-  ( ( y ( +g  `  G
) z )  e.  B  ->  ( (
x  e.  B  |->  ( M  .x.  x ) ) `  ( y ( +g  `  G
) z ) )  =  ( M  .x.  ( y ( +g  `  G ) z ) ) )
2117, 20syl 16 . . 3  |-  ( ( ( G  e.  Abel  /\  M  e.  ZZ )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( x  e.  B  |->  ( M  .x.  x ) ) `  ( y ( +g  `  G ) z ) )  =  ( M 
.x.  ( y ( +g  `  G ) z ) ) )
22 oveq2 6048 . . . . . 6  |-  ( x  =  y  ->  ( M  .x.  x )  =  ( M  .x.  y
) )
23 ovex 6065 . . . . . 6  |-  ( M 
.x.  y )  e. 
_V
2422, 9, 23fvmpt 5765 . . . . 5  |-  ( y  e.  B  ->  (
( x  e.  B  |->  ( M  .x.  x
) ) `  y
)  =  ( M 
.x.  y ) )
25 oveq2 6048 . . . . . 6  |-  ( x  =  z  ->  ( M  .x.  x )  =  ( M  .x.  z
) )
26 ovex 6065 . . . . . 6  |-  ( M 
.x.  z )  e. 
_V
2725, 9, 26fvmpt 5765 . . . . 5  |-  ( z  e.  B  ->  (
( x  e.  B  |->  ( M  .x.  x
) ) `  z
)  =  ( M 
.x.  z ) )
2824, 27oveqan12d 6059 . . . 4  |-  ( ( y  e.  B  /\  z  e.  B )  ->  ( ( ( x  e.  B  |->  ( M 
.x.  x ) ) `
 y ) ( +g  `  G ) ( ( x  e.  B  |->  ( M  .x.  x ) ) `  z ) )  =  ( ( M  .x.  y ) ( +g  `  G ) ( M 
.x.  z ) ) )
2928adantl 453 . . 3  |-  ( ( ( G  e.  Abel  /\  M  e.  ZZ )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( ( x  e.  B  |->  ( M 
.x.  x ) ) `
 y ) ( +g  `  G ) ( ( x  e.  B  |->  ( M  .x.  x ) ) `  z ) )  =  ( ( M  .x.  y ) ( +g  `  G ) ( M 
.x.  z ) ) )
3014, 21, 293eqtr4d 2446 . 2  |-  ( ( ( G  e.  Abel  /\  M  e.  ZZ )  /\  ( y  e.  B  /\  z  e.  B ) )  -> 
( ( x  e.  B  |->  ( M  .x.  x ) ) `  ( y ( +g  `  G ) z ) )  =  ( ( ( x  e.  B  |->  ( M  .x.  x
) ) `  y
) ( +g  `  G
) ( ( x  e.  B  |->  ( M 
.x.  x ) ) `
 z ) ) )
311, 1, 2, 2, 4, 4, 10, 30isghmd 14970 1  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  (
x  e.  B  |->  ( M  .x.  x ) )  e.  ( G 
GrpHom  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   ZZcz 10238   Basecbs 13424   +g cplusg 13484   Grpcgrp 14640  .gcmg 14644    GrpHom cghm 14958   Abelcabel 15368
This theorem is referenced by:  gsummulglem  15491
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-mulg 14770  df-ghm 14959  df-cmn 15369  df-abl 15370
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