MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulgrhm Structured version   Unicode version

Theorem mulgrhm 17826
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 17789 . 2  |-  ZZ  =  ( Base ` ring )
2 zring1 17794 . 2  |-  1  =  ( 1r ` ring )
3 mulgrhm.1 . 2  |-  .1.  =  ( 1r `  R )
4 zringmulr 17792 . 2  |-  x.  =  ( .r ` ring )
5 eqid 2441 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
6 zringrng 17786 . . 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 10672 . . . 4  |-  1  e.  ZZ
10 oveq1 6097 . . . . 5  |-  ( n  =  1  ->  (
n  .x.  .1.  )  =  ( 1  .x. 
.1.  ) )
11 mulgghm2.f . . . . 5  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
12 ovex 6115 . . . . 5  |-  ( 1 
.x.  .1.  )  e.  _V
1310, 11, 12fvmpt 5771 . . . 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, 3rngidcl 16655 . . . 4  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
17 mulgghm2.m . . . . 5  |-  .x.  =  (.g
`  R )
1815, 17mulg1 15627 . . . 4  |-  (  .1. 
e.  ( Base `  R
)  ->  ( 1 
.x.  .1.  )  =  .1.  )
1916, 18syl 16 . . 3  |-  ( R  e.  Ring  ->  ( 1 
.x.  .1.  )  =  .1.  )
2014, 19syl5eq 2485 . 2  |-  ( R  e.  Ring  ->  ( F `
 1 )  =  .1.  )
21 rnggrp 16640 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2221adantr 462 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Grp )
23 simprr 751 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
2416adantr 462 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  .1.  e.  ( Base `  R )
)
2515, 17mulgcl 15637 . . . . . . 7  |-  ( ( R  e.  Grp  /\  y  e.  ZZ  /\  .1.  e.  ( Base `  R
) )  ->  (
y  .x.  .1.  )  e.  ( Base `  R
) )
2622, 23, 24, 25syl3anc 1213 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( y  .x.  .1.  )  e.  (
Base `  R )
)
2715, 5, 3rnglidm 16658 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
y  .x.  .1.  )  e.  ( Base `  R
) )  ->  (  .1.  ( .r `  R
) ( y  .x.  .1.  ) )  =  ( y  .x.  .1.  )
)
2826, 27syldan 467 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  (  .1.  ( .r `  R ) ( y  .x.  .1.  ) )  =  ( y  .x.  .1.  )
)
2928oveq2d 6106 . . . 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 454 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Ring )
31 simprl 750 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
3215, 17, 5mulgass2 16682 . . . . 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 1215 . . . 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 15650 . . . . 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 1215 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (
x  x.  y ) 
.x.  .1.  )  =  ( x  .x.  ( y 
.x.  .1.  ) )
)
3629, 33, 353eqtr4rd 2484 . . 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 10689 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
3837adantl 463 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  x.  y )  e.  ZZ )
39 oveq1 6097 . . . . 5  |-  ( n  =  ( x  x.  y )  ->  (
n  .x.  .1.  )  =  ( ( x  x.  y )  .x.  .1.  ) )
40 ovex 6115 . . . . 5  |-  ( ( x  x.  y ) 
.x.  .1.  )  e.  _V
4139, 11, 40fvmpt 5771 . . . 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 6097 . . . . . 6  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
44 ovex 6115 . . . . . 6  |-  ( x 
.x.  .1.  )  e.  _V
4543, 11, 44fvmpt 5771 . . . . 5  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
46 oveq1 6097 . . . . . 6  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
47 ovex 6115 . . . . . 6  |-  ( y 
.x.  .1.  )  e.  _V
4846, 11, 47fvmpt 5771 . . . . 5  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
4945, 48oveqan12d 6109 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( F `  x ) ( .r
`  R ) ( F `  y ) )  =  ( ( x  .x.  .1.  )
( .r `  R
) ( y  .x.  .1.  ) ) )
5049adantl 463 . . 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 2483 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( F `  ( x  x.  y
) )  =  ( ( F `  x
) ( .r `  R ) ( F `
 y ) ) )
5217, 11, 15mulgghm2 17825 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  ( Base `  R
) )  ->  F  e.  (ring  GrpHom  R ) )
5321, 16, 52syl2anc 656 . 2  |-  ( R  e.  Ring  ->  F  e.  (ring  GrpHom  R ) )
541, 2, 3, 4, 5, 7, 8, 20, 51, 53isrhm2d 16806 1  |-  ( R  e.  Ring  ->  F  e.  (ring RingHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   1c1 9279    x. cmul 9283   ZZcz 10642   Basecbs 14170   .rcmulr 14235   Grpcgrp 15406  .gcmg 15410    GrpHom cghm 15737   1rcur 16593   Ringcrg 16635   RingHom crh 16794  ℤringzring 17783
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-seq 11803  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-mnd 15411  df-mhm 15460  df-grp 15538  df-minusg 15539  df-mulg 15541  df-subg 15671  df-ghm 15738  df-cmn 16272  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-rnghom 16796  df-subrg 16843  df-cnfld 17719  df-zring 17784
This theorem is referenced by:  mulgrhm2  17827
  Copyright terms: Public domain W3C validator