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Theorem mulgghm2 16741
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 444 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  R  e.  Grp )
2 zsubrg 16707 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 mulgghm2.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgrng 15826 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
5 rnggrp 15624 . . . 4  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
62, 4, 5mp2b 10 . . 3  |-  Z  e. 
Grp
71, 6jctil 524 . 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 14862 . . . . . 6  |-  ( ( R  e.  Grp  /\  n  e.  ZZ  /\  .1.  e.  B )  ->  (
n  .x.  .1.  )  e.  B )
11103expa 1153 . . . . 5  |-  ( ( ( R  e.  Grp  /\  n  e.  ZZ )  /\  .1.  e.  B
)  ->  ( n  .x.  .1.  )  e.  B
)
1211an32s 780 . . . 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 5852 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F : ZZ --> B )
15 eqid 2404 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
168, 9, 15mulgdir 14870 . . . . . . . 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 1171 . . . . . . 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 578 . . . . . 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 780 . . . . 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 10273 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  +  y )  e.  ZZ )
2120adantl 453 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  +  y )  e.  ZZ )
22 oveq1 6047 . . . . . . 7  |-  ( n  =  ( x  +  y )  ->  (
n  .x.  .1.  )  =  ( ( x  +  y )  .x.  .1.  ) )
23 ovex 6065 . . . . . . 7  |-  ( ( x  +  y ) 
.x.  .1.  )  e.  _V
2422, 13, 23fvmpt 5765 . . . . . 6  |-  ( ( x  +  y )  e.  ZZ  ->  ( F `  ( x  +  y ) )  =  ( ( x  +  y )  .x.  .1.  ) )
2521, 24syl 16 . . . . 5  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( F `  (
x  +  y ) )  =  ( ( x  +  y ) 
.x.  .1.  ) )
26 oveq1 6047 . . . . . . . 8  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
27 ovex 6065 . . . . . . . 8  |-  ( x 
.x.  .1.  )  e.  _V
2826, 13, 27fvmpt 5765 . . . . . . 7  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
29 oveq1 6047 . . . . . . . 8  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
30 ovex 6065 . . . . . . . 8  |-  ( y 
.x.  .1.  )  e.  _V
3129, 13, 30fvmpt 5765 . . . . . . 7  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
3228, 31oveqan12d 6059 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( F `  x ) ( +g  `  R ) ( F `
 y ) )  =  ( ( x 
.x.  .1.  ) ( +g  `  R ) ( y  .x.  .1.  )
) )
3332adantl 453 . . . . 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 2446 . . . 4  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( F `  (
x  +  y ) )  =  ( ( F `  x ) ( +g  `  R
) ( F `  y ) ) )
3534ralrimivva 2758 . . 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 519 . 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 15832 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
382, 37ax-mp 8 . . 3  |-  ZZ  =  ( Base `  Z )
39 cnfldadd 16663 . . . . 5  |-  +  =  ( +g  ` fld )
403, 39ressplusg 13526 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
412, 40ax-mp 8 . . 3  |-  +  =  ( +g  `  Z )
4238, 8, 41, 15isghm 14961 . 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 646 1  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F  e.  ( Z  GrpHom  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040    + caddc 8949   ZZcz 10238   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   Grpcgrp 14640  .gcmg 14644    GrpHom cghm 14958   Ringcrg 15615  SubRingcsubrg 15819  ℂfldccnfld 16658
This theorem is referenced by:  mulgrhm  16742  frgpcyg  16809
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  ax-addf 9025  ax-mulf 9026
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-int 4011  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-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  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-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-seq 11279  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-subrg 15821  df-cnfld 16659
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