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Theorem mulgghm2OLD 18054
Description: The powers of a group element give a homomorphism from 
ZZ to a group. (Contributed by Mario Carneiro, 13-Jun-2015.) Obsolete version of mulgghm2 18051 as of 12-Jun-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
mulgghm2OLD.1  |-  Z  =  (flds  ZZ )
mulgghm2OLD.2  |-  .x.  =  (.g
`  R )
mulgghm2OLD.3  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
mulgghm2OLD.4  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
mulgghm2OLD  |-  ( ( 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 mulgghm2OLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  R  e.  Grp )
2 zsubrg 17992 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 mulgghm2OLD.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgrng 16992 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
5 rnggrp 16774 . . . 4  |-  ( Z  e.  Ring  ->  Z  e. 
Grp )
62, 4, 5mp2b 10 . . 3  |-  Z  e. 
Grp
71, 6jctil 537 . 2  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  -> 
( Z  e.  Grp  /\  R  e.  Grp )
)
8 mulgghm2OLD.4 . . . . . . 7  |-  B  =  ( Base `  R
)
9 mulgghm2OLD.2 . . . . . . 7  |-  .x.  =  (.g
`  R )
108, 9mulgcl 15764 . . . . . 6  |-  ( ( R  e.  Grp  /\  n  e.  ZZ  /\  .1.  e.  B )  ->  (
n  .x.  .1.  )  e.  B )
11103expa 1188 . . . . 5  |-  ( ( ( R  e.  Grp  /\  n  e.  ZZ )  /\  .1.  e.  B
)  ->  ( n  .x.  .1.  )  e.  B
)
1211an32s 802 . . . 4  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  n  e.  ZZ )  ->  ( n  .x.  .1.  )  e.  B
)
13 mulgghm2OLD.3 . . . 4  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
1412, 13fmptd 5977 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F : ZZ --> B )
15 eqid 2454 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
168, 9, 15mulgdir 15772 . . . . . . . 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 1206 . . . . . . 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 594 . . . . . 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 802 . . . . 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 10797 . . . . . . 7  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  +  y )  e.  ZZ )
2120adantl 466 . . . . . 6  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( x  +  y )  e.  ZZ )
22 oveq1 6208 . . . . . . 7  |-  ( n  =  ( x  +  y )  ->  (
n  .x.  .1.  )  =  ( ( x  +  y )  .x.  .1.  ) )
23 ovex 6226 . . . . . . 7  |-  ( ( x  +  y ) 
.x.  .1.  )  e.  _V
2422, 13, 23fvmpt 5884 . . . . . 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 6208 . . . . . . . 8  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
27 ovex 6226 . . . . . . . 8  |-  ( x 
.x.  .1.  )  e.  _V
2826, 13, 27fvmpt 5884 . . . . . . 7  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
29 oveq1 6208 . . . . . . . 8  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
30 ovex 6226 . . . . . . . 8  |-  ( y 
.x.  .1.  )  e.  _V
3129, 13, 30fvmpt 5884 . . . . . . 7  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
3228, 31oveqan12d 6220 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( F `  x ) ( +g  `  R ) ( F `
 y ) )  =  ( ( x 
.x.  .1.  ) ( +g  `  R ) ( y  .x.  .1.  )
) )
3332adantl 466 . . . . 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 2505 . . . 4  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( F `  (
x  +  y ) )  =  ( ( F `  x ) ( +g  `  R
) ( F `  y ) ) )
3534ralrimivva 2914 . . 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 532 . 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 16998 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
382, 37ax-mp 5 . . 3  |-  ZZ  =  ( Base `  Z )
39 cnfldadd 17949 . . . . 5  |-  +  =  ( +g  ` fld )
403, 39ressplusg 14400 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
412, 40ax-mp 5 . . 3  |-  +  =  ( +g  `  Z )
4238, 8, 41, 15isghm 15867 . 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 664 1  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F  e.  ( Z  GrpHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    |-> cmpt 4459   -->wf 5523   ` cfv 5527  (class class class)co 6201    + caddc 9397   ZZcz 10758   Basecbs 14293   ↾s cress 14294   +g cplusg 14358   Grpcgrp 15530  .gcmg 15534    GrpHom cghm 15864   Ringcrg 16769  SubRingcsubrg 16985  ℂfldccnfld 17944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-addf 9473  ax-mulf 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-fz 11556  df-seq 11925  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-starv 14373  df-tset 14377  df-ple 14378  df-ds 14380  df-unif 14381  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666  df-mulg 15668  df-subg 15798  df-ghm 15865  df-cmn 16401  df-mgp 16715  df-ur 16727  df-rng 16771  df-cring 16772  df-subrg 16987  df-cnfld 17945
This theorem is referenced by:  mulgrhmOLD  18055
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