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Theorem mulgghm2OLD 18294
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 18291 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 18232 . . . 4  |-  ZZ  e.  (SubRing ` fld )
3 mulgghm2OLD.1 . . . . 5  |-  Z  =  (flds  ZZ )
43subrgrng 17208 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
5 rnggrp 16984 . . . 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 15952 . . . . . 6  |-  ( ( R  e.  Grp  /\  n  e.  ZZ  /\  .1.  e.  B )  ->  (
n  .x.  .1.  )  e.  B )
11103expa 1191 . . . . 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 6036 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  B )  ->  F : ZZ --> B )
15 eqid 2460 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
168, 9, 15mulgdir 15960 . . . . . . . 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 1209 . . . . . . 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 10892 . . . . . . 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 6282 . . . . . . 7  |-  ( n  =  ( x  +  y )  ->  (
n  .x.  .1.  )  =  ( ( x  +  y )  .x.  .1.  ) )
23 ovex 6300 . . . . . . 7  |-  ( ( x  +  y ) 
.x.  .1.  )  e.  _V
2422, 13, 23fvmpt 5941 . . . . . 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 6282 . . . . . . . 8  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
27 ovex 6300 . . . . . . . 8  |-  ( x 
.x.  .1.  )  e.  _V
2826, 13, 27fvmpt 5941 . . . . . . 7  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
29 oveq1 6282 . . . . . . . 8  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
30 ovex 6300 . . . . . . . 8  |-  ( y 
.x.  .1.  )  e.  _V
3129, 13, 30fvmpt 5941 . . . . . . 7  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
3228, 31oveqan12d 6294 . . . . . 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 2511 . . . 4  |-  ( ( ( R  e.  Grp  /\  .1.  e.  B )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  -> 
( F `  (
x  +  y ) )  =  ( ( F `  x ) ( +g  `  R
) ( F `  y ) ) )
3534ralrimivva 2878 . . 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 17214 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
382, 37ax-mp 5 . . 3  |-  ZZ  =  ( Base `  Z )
39 cnfldadd 18189 . . . . 5  |-  +  =  ( +g  ` fld )
403, 39ressplusg 14586 . . . 4  |-  ( ZZ  e.  (SubRing ` fld )  ->  +  =  ( +g  `  Z ) )
412, 40ax-mp 5 . . 3  |-  +  =  ( +g  `  Z )
4238, 8, 41, 15isghm 16055 . 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 1374    e. wcel 1762   A.wral 2807    |-> cmpt 4498   -->wf 5575   ` cfv 5579  (class class class)co 6275    + caddc 9484   ZZcz 10853   Basecbs 14479   ↾s cress 14480   +g cplusg 14544   Grpcgrp 15716  .gcmg 15720    GrpHom cghm 16052   Ringcrg 16979  SubRingcsubrg 17201  ℂfldccnfld 18184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-seq 12064  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-mulg 15854  df-subg 15986  df-ghm 16053  df-cmn 16589  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-subrg 17203  df-cnfld 18185
This theorem is referenced by:  mulgrhmOLD  18295
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