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Theorem mulgrhmOLD 18049
Description: The powers of the element  1 give a ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) Obsolete version of mulgrhm 18046 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.  )
)
mulgrhmOLD.4  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mulgrhmOLD  |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R )
)
Distinct variable groups:    R, n    .x. ,
n    n, Z    .1. , n
Allowed substitution hint:    F( n)

Proof of Theorem mulgrhmOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zsubrg 17986 . . 3  |-  ZZ  e.  (SubRing ` fld )
2 mulgghm2OLD.1 . . . 4  |-  Z  =  (flds  ZZ )
32subrgbas 16992 . . 3  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
41, 3ax-mp 5 . 2  |-  ZZ  =  ( Base `  Z )
5 cnfld1 17961 . . . 4  |-  1  =  ( 1r ` fld )
62, 5subrg1 16993 . . 3  |-  ( ZZ  e.  (SubRing ` fld )  ->  1  =  ( 1r `  Z
) )
71, 6ax-mp 5 . 2  |-  1  =  ( 1r `  Z )
8 mulgrhmOLD.4 . 2  |-  .1.  =  ( 1r `  R )
9 cnfldmul 17944 . . . 4  |-  x.  =  ( .r ` fld )
102, 9ressmulr 14405 . . 3  |-  ( ZZ  e.  (SubRing ` fld )  ->  x.  =  ( .r `  Z ) )
111, 10ax-mp 5 . 2  |-  x.  =  ( .r `  Z )
12 eqid 2452 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
132subrgrng 16986 . . 3  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
141, 13mp1i 12 . 2  |-  ( R  e.  Ring  ->  Z  e. 
Ring )
15 id 22 . 2  |-  ( R  e.  Ring  ->  R  e. 
Ring )
16 1z 10782 . . . 4  |-  1  e.  ZZ
17 oveq1 6202 . . . . 5  |-  ( n  =  1  ->  (
n  .x.  .1.  )  =  ( 1  .x. 
.1.  ) )
18 mulgghm2OLD.3 . . . . 5  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
19 ovex 6220 . . . . 5  |-  ( 1 
.x.  .1.  )  e.  _V
2017, 18, 19fvmpt 5878 . . . 4  |-  ( 1  e.  ZZ  ->  ( F `  1 )  =  ( 1  .x. 
.1.  ) )
2116, 20ax-mp 5 . . 3  |-  ( F `
 1 )  =  ( 1  .x.  .1.  )
22 eqid 2452 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
2322, 8rngidcl 16783 . . . 4  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
24 mulgghm2OLD.2 . . . . 5  |-  .x.  =  (.g
`  R )
2522, 24mulg1 15748 . . . 4  |-  (  .1. 
e.  ( Base `  R
)  ->  ( 1 
.x.  .1.  )  =  .1.  )
2623, 25syl 16 . . 3  |-  ( R  e.  Ring  ->  ( 1 
.x.  .1.  )  =  .1.  )
2721, 26syl5eq 2505 . 2  |-  ( R  e.  Ring  ->  ( F `
 1 )  =  .1.  )
28 rnggrp 16768 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2928adantr 465 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Grp )
30 simprr 756 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
3123adantr 465 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  .1.  e.  ( Base `  R )
)
3222, 24mulgcl 15758 . . . . . . 7  |-  ( ( R  e.  Grp  /\  y  e.  ZZ  /\  .1.  e.  ( Base `  R
) )  ->  (
y  .x.  .1.  )  e.  ( Base `  R
) )
3329, 30, 31, 32syl3anc 1219 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( y  .x.  .1.  )  e.  (
Base `  R )
)
3422, 12, 8rnglidm 16786 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
y  .x.  .1.  )  e.  ( Base `  R
) )  ->  (  .1.  ( .r `  R
) ( y  .x.  .1.  ) )  =  ( y  .x.  .1.  )
)
3533, 34syldan 470 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  (  .1.  ( .r `  R ) ( y  .x.  .1.  ) )  =  ( y  .x.  .1.  )
)
3635oveq2d 6211 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  .x.  (  .1.  ( .r `  R ) ( y  .x.  .1.  )
) )  =  ( x  .x.  ( y 
.x.  .1.  ) )
)
37 simpl 457 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Ring )
38 simprl 755 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
3922, 24, 12mulgass2 16810 . . . . 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.  ) )
) )
4037, 38, 31, 33, 39syl13anc 1221 . . . 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.  ) ) ) )
4122, 24mulgass 15771 . . . . 5  |-  ( ( R  e.  Grp  /\  ( x  e.  ZZ  /\  y  e.  ZZ  /\  .1.  e.  ( Base `  R
) ) )  -> 
( ( x  x.  y )  .x.  .1.  )  =  ( x  .x.  ( y  .x.  .1.  ) ) )
4229, 38, 30, 31, 41syl13anc 1221 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (
x  x.  y ) 
.x.  .1.  )  =  ( x  .x.  ( y 
.x.  .1.  ) )
)
4336, 40, 423eqtr4rd 2504 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (
x  x.  y ) 
.x.  .1.  )  =  ( ( x  .x.  .1.  ) ( .r `  R ) ( y 
.x.  .1.  ) )
)
44 zmulcl 10799 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
4544adantl 466 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  x.  y )  e.  ZZ )
46 oveq1 6202 . . . . 5  |-  ( n  =  ( x  x.  y )  ->  (
n  .x.  .1.  )  =  ( ( x  x.  y )  .x.  .1.  ) )
47 ovex 6220 . . . . 5  |-  ( ( x  x.  y ) 
.x.  .1.  )  e.  _V
4846, 18, 47fvmpt 5878 . . . 4  |-  ( ( x  x.  y )  e.  ZZ  ->  ( F `  ( x  x.  y ) )  =  ( ( x  x.  y )  .x.  .1.  ) )
4945, 48syl 16 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( F `  ( x  x.  y
) )  =  ( ( x  x.  y
)  .x.  .1.  )
)
50 oveq1 6202 . . . . . 6  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
51 ovex 6220 . . . . . 6  |-  ( x 
.x.  .1.  )  e.  _V
5250, 18, 51fvmpt 5878 . . . . 5  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
53 oveq1 6202 . . . . . 6  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
54 ovex 6220 . . . . . 6  |-  ( y 
.x.  .1.  )  e.  _V
5553, 18, 54fvmpt 5878 . . . . 5  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
5652, 55oveqan12d 6214 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( F `  x ) ( .r
`  R ) ( F `  y ) )  =  ( ( x  .x.  .1.  )
( .r `  R
) ( y  .x.  .1.  ) ) )
5756adantl 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.  ) )
)
5843, 49, 573eqtr4d 2503 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( F `  ( x  x.  y
) )  =  ( ( F `  x
) ( .r `  R ) ( F `
 y ) ) )
592, 24, 18, 22mulgghm2OLD 18048 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  ( Base `  R
) )  ->  F  e.  ( Z  GrpHom  R ) )
6028, 23, 59syl2anc 661 . 2  |-  ( R  e.  Ring  ->  F  e.  ( Z  GrpHom  R ) )
614, 7, 8, 11, 12, 14, 15, 27, 58, 60isrhm2d 16936 1  |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195   1c1 9389    x. cmul 9393   ZZcz 10752   Basecbs 14287   ↾s cress 14288   .rcmulr 14353   Grpcgrp 15524  .gcmg 15528    GrpHom cghm 15858   1rcur 16720   Ringcrg 16763   RingHom crh 16922  SubRingcsubrg 16979  ℂfldccnfld 17938
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 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-addf 9467  ax-mulf 9468
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-fz 11550  df-seq 11919  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-0g 14494  df-mnd 15529  df-mhm 15578  df-grp 15659  df-minusg 15660  df-mulg 15662  df-subg 15792  df-ghm 15859  df-cmn 16395  df-mgp 16709  df-ur 16721  df-rng 16765  df-cring 16766  df-rnghom 16924  df-subrg 16981  df-cnfld 17939
This theorem is referenced by:  mulgrhm2OLD  18050
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