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Theorem mulgghm2 16459
Description: The powers of a group element give a homomorphism from 
ZZ to a group. (Contributed by Mario Carneiro, 13-Jun-2015.)
Hypotheses
Ref Expression
mulgghm2.1  |-  Z  =  (flds  ZZ )
mulgghm2.2  |-  .x.  =  (.g
`  R )
mulgghm2.3  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
mulgghm2.4  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
mulgghm2  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F  e.  ( Z  GrpHom  R ) )
Distinct variable groups:    B, n    R, n    .x. , n    n, Z    .1. ,
n
Allowed substitution hint:    F( n)

Proof of Theorem mulgghm2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  R  e.  Grp )
2 zsubrg 16425 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 mulgghm2.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgrng 15548 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
5 rnggrp 15346 . . . 4  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
62, 4, 5mp2b 9 . . 3  |-  Z  e. 
Grp
71, 6jctil 523 . 2  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( Z  e.  Grp  /\  R  e.  Grp )
)
8 mulgghm2.4 . . . . . . 7  |-  B  =  ( Base `  R
)
9 mulgghm2.2 . . . . . . 7  |-  .x.  =  (.g
`  R )
108, 9mulgcl 14584 . . . . . 6  |-  ( ( R  e.  Grp  /\  n  e.  ZZ  /\  .1.  e.  B )  ->  (
n  .x.  .1.  )  e.  B )
11103expa 1151 . . . . 5  |-  ( ( ( R  e.  Grp  /\  n  e.  ZZ )  /\  .1.  e.  B
)  ->  ( n  .x.  .1.  )  e.  B
)
1211an32s 779 . . . 4  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  n  e.  ZZ )  ->  ( n  .x.  .1.  )  e.  B
)
13 mulgghm2.3 . . . 4  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
1412, 13fmptd 5684 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F : ZZ --> B )
15 eqid 2283 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
168, 9, 15mulgdir 14592 . . . . . . . 8  |-  ( ( R  e.  Grp  /\  ( x  e.  ZZ  /\  y  e.  ZZ  /\  .1.  e.  B ) )  ->  ( ( x  +  y )  .x.  .1.  )  =  (
( x  .x.  .1.  ) ( +g  `  R
) ( y  .x.  .1.  ) ) )
17163exp2 1169 . . . . . . 7  |-  ( R  e.  Grp  ->  (
x  e.  ZZ  ->  ( y  e.  ZZ  ->  (  .1.  e.  B  -> 
( ( x  +  y )  .x.  .1.  )  =  ( (
x  .x.  .1.  )
( +g  `  R ) ( y  .x.  .1.  ) ) ) ) ) )
1817imp42 577 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  /\  .1.  e.  B )  ->  (
( x  +  y )  .x.  .1.  )  =  ( ( x 
.x.  .1.  ) ( +g  `  R ) ( y  .x.  .1.  )
) )
1918an32s 779 . . . . 5  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( x  +  y )  .x.  .1.  )  =  ( (
x  .x.  .1.  )
( +g  `  R ) ( y  .x.  .1.  ) ) )
20 zaddcl 10059 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  +  y )  e.  ZZ )
2120adantl 452 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  +  y )  e.  ZZ )
22 oveq1 5865 . . . . . . 7  |-  ( n  =  ( x  +  y )  ->  (
n  .x.  .1.  )  =  ( ( x  +  y )  .x.  .1.  ) )
23 ovex 5883 . . . . . . 7  |-  ( ( x  +  y ) 
.x.  .1.  )  e.  _V
2422, 13, 23fvmpt 5602 . . . . . 6  |-  ( ( x  +  y )  e.  ZZ  ->  ( F `  ( x  +  y ) )  =  ( ( x  +  y )  .x.  .1.  ) )
2521, 24syl 15 . . . . 5  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( F `  (
x  +  y ) )  =  ( ( x  +  y ) 
.x.  .1.  ) )
26 oveq1 5865 . . . . . . . 8  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
27 ovex 5883 . . . . . . . 8  |-  ( x 
.x.  .1.  )  e.  _V
2826, 13, 27fvmpt 5602 . . . . . . 7  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
29 oveq1 5865 . . . . . . . 8  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
30 ovex 5883 . . . . . . . 8  |-  ( y 
.x.  .1.  )  e.  _V
3129, 13, 30fvmpt 5602 . . . . . . 7  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
3228, 31oveqan12d 5877 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( F `  x ) ( +g  `  R ) ( F `
 y ) )  =  ( ( x 
.x.  .1.  ) ( +g  `  R ) ( y  .x.  .1.  )
) )
3332adantl 452 . . . . 5  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( ( F `  x ) ( +g  `  R ) ( F `
 y ) )  =  ( ( x 
.x.  .1.  ) ( +g  `  R ) ( y  .x.  .1.  )
) )
3419, 25, 333eqtr4d 2325 . . . 4  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( F `  (
x  +  y ) )  =  ( ( F `  x ) ( +g  `  R
) ( F `  y ) ) )
3534ralrimivva 2635 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  A. x  e.  ZZ  A. y  e.  ZZ  ( F `  ( x  +  y ) )  =  ( ( F `
 x ) ( +g  `  R ) ( F `  y
) ) )
3614, 35jca 518 . 2  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( F : ZZ --> B  /\  A. x  e.  ZZ  A. y  e.  ZZ  ( F `  ( x  +  y
) )  =  ( ( F `  x
) ( +g  `  R
) ( F `  y ) ) ) )
373subrgbas 15554 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
382, 37ax-mp 8 . . 3  |-  ZZ  =  ( Base `  Z )
39 cnfldadd 16384 . . . . 5  |-  +  =  ( +g  ` fld )
403, 39ressplusg 13250 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
412, 40ax-mp 8 . . 3  |-  +  =  ( +g  `  Z )
4238, 8, 41, 15isghm 14683 . 2  |-  ( F  e.  ( Z  GrpHom  R )  <->  ( ( Z  e.  Grp  /\  R  e.  Grp )  /\  ( F : ZZ --> B  /\  A. x  e.  ZZ  A. y  e.  ZZ  ( F `  ( x  +  y ) )  =  ( ( F `
 x ) ( +g  `  R ) ( F `  y
) ) ) ) )
437, 36, 42sylanbrc 645 1  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F  e.  ( Z  GrpHom  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    + caddc 8740   ZZcz 10024   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   Grpcgrp 14362  .gcmg 14366    GrpHom cghm 14680   Ringcrg 15337  SubRingcsubrg 15541  ℂfldccnfld 16377
This theorem is referenced by:  mulgrhm  16460  frgpcyg  16527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-seq 11047  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-subrg 15543  df-cnfld 16378
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