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Theorem mulgghm2 16709
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 16675 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 mulgghm2.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgrng 15798 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
5 rnggrp 15596 . . . 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 14834 . . . . . 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 5832 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F : ZZ --> B )
15 eqid 2387 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
168, 9, 15mulgdir 14842 . . . . . . . 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 10249 . . . . . . 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 6027 . . . . . . 7  |-  ( n  =  ( x  +  y )  ->  (
n  .x.  .1.  )  =  ( ( x  +  y )  .x.  .1.  ) )
23 ovex 6045 . . . . . . 7  |-  ( ( x  +  y ) 
.x.  .1.  )  e.  _V
2422, 13, 23fvmpt 5745 . . . . . 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 6027 . . . . . . . 8  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
27 ovex 6045 . . . . . . . 8  |-  ( x 
.x.  .1.  )  e.  _V
2826, 13, 27fvmpt 5745 . . . . . . 7  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
29 oveq1 6027 . . . . . . . 8  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
30 ovex 6045 . . . . . . . 8  |-  ( y 
.x.  .1.  )  e.  _V
3129, 13, 30fvmpt 5745 . . . . . . 7  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
3228, 31oveqan12d 6039 . . . . . 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 2429 . . . 4  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( F `  (
x  +  y ) )  =  ( ( F `  x ) ( +g  `  R
) ( F `  y ) ) )
3534ralrimivva 2741 . . 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 15804 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
382, 37ax-mp 8 . . 3  |-  ZZ  =  ( Base `  Z )
39 cnfldadd 16631 . . . . 5  |-  +  =  ( +g  ` fld )
403, 39ressplusg 13498 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
412, 40ax-mp 8 . . 3  |-  +  =  ( +g  `  Z )
4238, 8, 41, 15isghm 14933 . 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 1717   A.wral 2649    e. cmpt 4207   -->wf 5390   ` cfv 5394  (class class class)co 6020    + caddc 8926   ZZcz 10214   Basecbs 13396   ↾s cress 13397   +g cplusg 13456   Grpcgrp 14612  .gcmg 14616    GrpHom cghm 14930   Ringcrg 15587  SubRingcsubrg 15791  ℂfldccnfld 16626
This theorem is referenced by:  mulgrhm  16710  frgpcyg  16777
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-seq 11251  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-mulg 14742  df-subg 14868  df-ghm 14931  df-cmn 15341  df-mgp 15576  df-rng 15590  df-cring 15591  df-ur 15592  df-subrg 15793  df-cnfld 16627
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