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Theorem mulgrhm 18294
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 18257 . 2  |-  ZZ  =  ( Base ` ring )
2 zring1 18262 . 2  |-  1  =  ( 1r ` ring )
3 mulgrhm.1 . 2  |-  .1.  =  ( 1r `  R )
4 zringmulr 18260 . 2  |-  x.  =  ( .r ` ring )
5 eqid 2462 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
6 zringrng 18254 . . 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 10885 . . . 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 6302 . . . . 5  |-  ( 1 
.x.  .1.  )  e.  _V
1310, 11, 12fvmpt 5943 . . . 4  |-  ( 1  e.  ZZ  ->  ( F `  1 )  =  ( 1  .x. 
.1.  ) )
149, 13ax-mp 5 . . 3  |-  ( F `
 1 )  =  ( 1  .x.  .1.  )
15 eqid 2462 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
1615, 3rngidcl 17001 . . . 4  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
17 mulgghm2.m . . . . 5  |-  .x.  =  (.g
`  R )
1815, 17mulg1 15944 . . . 4  |-  (  .1. 
e.  ( Base `  R
)  ->  ( 1 
.x.  .1.  )  =  .1.  )
1916, 18syl 16 . . 3  |-  ( R  e.  Ring  ->  ( 1 
.x.  .1.  )  =  .1.  )
2014, 19syl5eq 2515 . 2  |-  ( R  e.  Ring  ->  ( F `
 1 )  =  .1.  )
21 rnggrp 16986 . . . . . . . 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 15954 . . . . . . 7  |-  ( ( R  e.  Grp  /\  y  e.  ZZ  /\  .1.  e.  ( Base `  R
) )  ->  (
y  .x.  .1.  )  e.  ( Base `  R
) )
2622, 23, 24, 25syl3anc 1223 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( y  .x.  .1.  )  e.  (
Base `  R )
)
2715, 5, 3rnglidm 17004 . . . . . 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 17028 . . . . 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 1225 . . . 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 15967 . . . . 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 1225 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (
x  x.  y ) 
.x.  .1.  )  =  ( x  .x.  ( y 
.x.  .1.  ) )
)
3629, 33, 353eqtr4rd 2514 . . 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 10902 . . . . 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 6302 . . . . 5  |-  ( ( x  x.  y ) 
.x.  .1.  )  e.  _V
4139, 11, 40fvmpt 5943 . . . 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 6302 . . . . . 6  |-  ( x 
.x.  .1.  )  e.  _V
4543, 11, 44fvmpt 5943 . . . . 5  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
46 oveq1 6284 . . . . . 6  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
47 ovex 6302 . . . . . 6  |-  ( y 
.x.  .1.  )  e.  _V
4846, 11, 47fvmpt 5943 . . . . 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 2513 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( F `  ( x  x.  y
) )  =  ( ( F `  x
) ( .r `  R ) ( F `
 y ) ) )
5217, 11, 15mulgghm2 18293 . . 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 17156 1  |-  ( R  e.  Ring  ->  F  e.  (ring RingHom  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277   1c1 9484    x. cmul 9488   ZZcz 10855   Basecbs 14481   .rcmulr 14547   Grpcgrp 15718  .gcmg 15722    GrpHom cghm 16054   1rcur 16938   Ringcrg 16981   RingHom crh 17140  ℤringzring 18251
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-addf 9562  ax-mulf 9563
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-seq 12066  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-0g 14688  df-mnd 15723  df-mhm 15772  df-grp 15853  df-minusg 15854  df-mulg 15856  df-subg 15988  df-ghm 16055  df-cmn 16591  df-mgp 16927  df-ur 16939  df-rng 16983  df-cring 16984  df-rnghom 17143  df-subrg 17205  df-cnfld 18187  df-zring 18252
This theorem is referenced by:  mulgrhm2  18295
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