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Theorem mulgrhm 18399
Description: The powers of the element  1 give a ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
Hypotheses
Ref Expression
mulgghm2.m  |-  .x.  =  (.g
`  R )
mulgghm2.f  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
mulgrhm.1  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mulgrhm  |-  ( R  e.  Ring  ->  F  e.  (ring RingHom  R ) )
Distinct variable groups:    R, n    .x. ,
n    .1. , n
Allowed substitution hint:    F( n)

Proof of Theorem mulgrhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zringbas 18362 . 2  |-  ZZ  =  ( Base ` ring )
2 zring1 18367 . 2  |-  1  =  ( 1r ` ring )
3 mulgrhm.1 . 2  |-  .1.  =  ( 1r `  R )
4 zringmulr 18365 . 2  |-  x.  =  ( .r ` ring )
5 eqid 2441 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
6 zringring 18359 . . 3  |-ring  e.  Ring
76a1i 11 . 2  |-  ( R  e.  Ring  ->ring  e.  Ring )
8 id 22 . 2  |-  ( R  e.  Ring  ->  R  e. 
Ring )
9 1z 10895 . . . 4  |-  1  e.  ZZ
10 oveq1 6284 . . . . 5  |-  ( n  =  1  ->  (
n  .x.  .1.  )  =  ( 1  .x. 
.1.  ) )
11 mulgghm2.f . . . . 5  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
12 ovex 6305 . . . . 5  |-  ( 1 
.x.  .1.  )  e.  _V
1310, 11, 12fvmpt 5937 . . . 4  |-  ( 1  e.  ZZ  ->  ( F `  1 )  =  ( 1  .x. 
.1.  ) )
149, 13ax-mp 5 . . 3  |-  ( F `
 1 )  =  ( 1  .x.  .1.  )
15 eqid 2441 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1615, 3ringidcl 17087 . . . 4  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
17 mulgghm2.m . . . . 5  |-  .x.  =  (.g
`  R )
1815, 17mulg1 16018 . . . 4  |-  (  .1. 
e.  ( Base `  R
)  ->  ( 1 
.x.  .1.  )  =  .1.  )
1916, 18syl 16 . . 3  |-  ( R  e.  Ring  ->  ( 1 
.x.  .1.  )  =  .1.  )
2014, 19syl5eq 2494 . 2  |-  ( R  e.  Ring  ->  ( F `
 1 )  =  .1.  )
21 ringgrp 17071 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2221adantr 465 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Grp )
23 simprr 756 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
2416adantr 465 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  .1.  e.  ( Base `  R )
)
2515, 17mulgcl 16028 . . . . . . 7  |-  ( ( R  e.  Grp  /\  y  e.  ZZ  /\  .1.  e.  ( Base `  R
) )  ->  (
y  .x.  .1.  )  e.  ( Base `  R
) )
2622, 23, 24, 25syl3anc 1227 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( y  .x.  .1.  )  e.  (
Base `  R )
)
2715, 5, 3ringlidm 17090 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
y  .x.  .1.  )  e.  ( Base `  R
) )  ->  (  .1.  ( .r `  R
) ( y  .x.  .1.  ) )  =  ( y  .x.  .1.  )
)
2826, 27syldan 470 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  (  .1.  ( .r `  R ) ( y  .x.  .1.  ) )  =  ( y  .x.  .1.  )
)
2928oveq2d 6293 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  .x.  (  .1.  ( .r `  R ) ( y  .x.  .1.  )
) )  =  ( x  .x.  ( y 
.x.  .1.  ) )
)
30 simpl 457 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Ring )
31 simprl 755 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
3215, 17, 5mulgass2 17115 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  .1.  e.  ( Base `  R
)  /\  ( y  .x.  .1.  )  e.  (
Base `  R )
) )  ->  (
( x  .x.  .1.  ) ( .r `  R ) ( y 
.x.  .1.  ) )  =  ( x  .x.  (  .1.  ( .r `  R ) ( y 
.x.  .1.  ) )
) )
3330, 31, 24, 26, 32syl13anc 1229 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (
x  .x.  .1.  )
( .r `  R
) ( y  .x.  .1.  ) )  =  ( x  .x.  (  .1.  ( .r `  R
) ( y  .x.  .1.  ) ) ) )
3415, 17mulgass 16041 . . . . 5  |-  ( ( R  e.  Grp  /\  ( x  e.  ZZ  /\  y  e.  ZZ  /\  .1.  e.  ( Base `  R
) ) )  -> 
( ( x  x.  y )  .x.  .1.  )  =  ( x  .x.  ( y  .x.  .1.  ) ) )
3522, 31, 23, 24, 34syl13anc 1229 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (
x  x.  y ) 
.x.  .1.  )  =  ( x  .x.  ( y 
.x.  .1.  ) )
)
3629, 33, 353eqtr4rd 2493 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (
x  x.  y ) 
.x.  .1.  )  =  ( ( x  .x.  .1.  ) ( .r `  R ) ( y 
.x.  .1.  ) )
)
37 zmulcl 10913 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
3837adantl 466 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  x.  y )  e.  ZZ )
39 oveq1 6284 . . . . 5  |-  ( n  =  ( x  x.  y )  ->  (
n  .x.  .1.  )  =  ( ( x  x.  y )  .x.  .1.  ) )
40 ovex 6305 . . . . 5  |-  ( ( x  x.  y ) 
.x.  .1.  )  e.  _V
4139, 11, 40fvmpt 5937 . . . 4  |-  ( ( x  x.  y )  e.  ZZ  ->  ( F `  ( x  x.  y ) )  =  ( ( x  x.  y )  .x.  .1.  ) )
4238, 41syl 16 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( F `  ( x  x.  y
) )  =  ( ( x  x.  y
)  .x.  .1.  )
)
43 oveq1 6284 . . . . . 6  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
44 ovex 6305 . . . . . 6  |-  ( x 
.x.  .1.  )  e.  _V
4543, 11, 44fvmpt 5937 . . . . 5  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
46 oveq1 6284 . . . . . 6  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
47 ovex 6305 . . . . . 6  |-  ( y 
.x.  .1.  )  e.  _V
4846, 11, 47fvmpt 5937 . . . . 5  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
4945, 48oveqan12d 6296 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( F `  x ) ( .r
`  R ) ( F `  y ) )  =  ( ( x  .x.  .1.  )
( .r `  R
) ( y  .x.  .1.  ) ) )
5049adantl 466 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( ( F `  x )
( .r `  R
) ( F `  y ) )  =  ( ( x  .x.  .1.  ) ( .r `  R ) ( y 
.x.  .1.  ) )
)
5136, 42, 503eqtr4d 2492 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( F `  ( x  x.  y
) )  =  ( ( F `  x
) ( .r `  R ) ( F `
 y ) ) )
5217, 11, 15mulgghm2 18398 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  ( Base `  R
) )  ->  F  e.  (ring  GrpHom  R ) )
5321, 16, 52syl2anc 661 . 2  |-  ( R  e.  Ring  ->  F  e.  (ring  GrpHom  R ) )
541, 2, 3, 4, 5, 7, 8, 20, 51, 53isrhm2d 17245 1  |-  ( R  e.  Ring  ->  F  e.  (ring RingHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    |-> cmpt 4491   ` cfv 5574  (class class class)co 6277   1c1 9491    x. cmul 9495   ZZcz 10865   Basecbs 14504   .rcmulr 14570   Grpcgrp 15922  .gcmg 15925    GrpHom cghm 16133   1rcur 17021   Ringcrg 17066   RingHom crh 17229  ℤringzring 18356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-fz 11677  df-seq 12082  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-mhm 15835  df-grp 15926  df-minusg 15927  df-mulg 15929  df-subg 16067  df-ghm 16134  df-cmn 16669  df-mgp 17010  df-ur 17022  df-ring 17068  df-cring 17069  df-rnghom 17232  df-subrg 17295  df-cnfld 18289  df-zring 18357
This theorem is referenced by:  mulgrhm2  18400
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