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Theorem mulgrhm 16788
Description: The powers of the element  1 give a ring homomorphism from  ZZ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
mulgghm2.1  |-  Z  =  (flds  ZZ )
mulgghm2.2  |-  .x.  =  (.g
`  R )
mulgghm2.3  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
mulgrhm.4  |-  .1.  =  ( 1r `  R )
Assertion
Ref Expression
mulgrhm  |-  ( 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 mulgrhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zsubrg 16753 . . 3  |-  ZZ  e.  (SubRing ` fld )
2 mulgghm2.1 . . . 4  |-  Z  =  (flds  ZZ )
32subrgbas 15878 . . 3  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  Z )
)
41, 3ax-mp 8 . 2  |-  ZZ  =  ( Base `  Z )
5 cnfld1 16727 . . . 4  |-  1  =  ( 1r ` fld )
62, 5subrg1 15879 . . 3  |-  ( ZZ  e.  (SubRing ` fld )  ->  1  =  ( 1r `  Z
) )
71, 6ax-mp 8 . 2  |-  1  =  ( 1r `  Z )
8 mulgrhm.4 . 2  |-  .1.  =  ( 1r `  R )
9 cnfldmul 16710 . . . 4  |-  x.  =  ( .r ` fld )
102, 9ressmulr 13583 . . 3  |-  ( ZZ  e.  (SubRing ` fld )  ->  x.  =  ( .r `  Z ) )
111, 10ax-mp 8 . 2  |-  x.  =  ( .r `  Z )
12 eqid 2437 . 2  |-  ( .r
`  R )  =  ( .r `  R
)
132subrgrng 15872 . . 3  |-  ( ZZ  e.  (SubRing ` fld )  ->  Z  e. 
Ring )
141, 13mp1i 12 . 2  |-  ( R  e.  Ring  ->  Z  e. 
Ring )
15 id 21 . 2  |-  ( R  e.  Ring  ->  R  e. 
Ring )
16 1z 10312 . . . 4  |-  1  e.  ZZ
17 oveq1 6089 . . . . 5  |-  ( n  =  1  ->  (
n  .x.  .1.  )  =  ( 1  .x. 
.1.  ) )
18 mulgghm2.3 . . . . 5  |-  F  =  ( n  e.  ZZ  |->  ( n  .x.  .1.  )
)
19 ovex 6107 . . . . 5  |-  ( 1 
.x.  .1.  )  e.  _V
2017, 18, 19fvmpt 5807 . . . 4  |-  ( 1  e.  ZZ  ->  ( F `  1 )  =  ( 1  .x. 
.1.  ) )
2116, 20ax-mp 8 . . 3  |-  ( F `
 1 )  =  ( 1  .x.  .1.  )
22 eqid 2437 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
2322, 8rngidcl 15685 . . . 4  |-  ( R  e.  Ring  ->  .1.  e.  ( Base `  R )
)
24 mulgghm2.2 . . . . 5  |-  .x.  =  (.g
`  R )
2522, 24mulg1 14898 . . . 4  |-  (  .1. 
e.  ( Base `  R
)  ->  ( 1 
.x.  .1.  )  =  .1.  )
2623, 25syl 16 . . 3  |-  ( R  e.  Ring  ->  ( 1 
.x.  .1.  )  =  .1.  )
2721, 26syl5eq 2481 . 2  |-  ( R  e.  Ring  ->  ( F `
 1 )  =  .1.  )
28 rnggrp 15670 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2928adantr 453 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Grp )
30 simprr 735 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  y  e.  ZZ )
3123adantr 453 . . . . . . 7  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  .1.  e.  ( Base `  R )
)
3222, 24mulgcl 14908 . . . . . . 7  |-  ( ( R  e.  Grp  /\  y  e.  ZZ  /\  .1.  e.  ( Base `  R
) )  ->  (
y  .x.  .1.  )  e.  ( Base `  R
) )
3329, 30, 31, 32syl3anc 1185 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( y  .x.  .1.  )  e.  (
Base `  R )
)
3422, 12, 8rnglidm 15688 . . . . . 6  |-  ( ( R  e.  Ring  /\  (
y  .x.  .1.  )  e.  ( Base `  R
) )  ->  (  .1.  ( .r `  R
) ( y  .x.  .1.  ) )  =  ( y  .x.  .1.  )
)
3533, 34syldan 458 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  (  .1.  ( .r `  R ) ( y  .x.  .1.  ) )  =  ( y  .x.  .1.  )
)
3635oveq2d 6098 . . . 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 445 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  R  e.  Ring )
38 simprl 734 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  x  e.  ZZ )
3922, 24, 12mulgass2 15711 . . . . 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 1187 . . . 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 14921 . . . . 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 1187 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( (
x  x.  y ) 
.x.  .1.  )  =  ( x  .x.  ( y 
.x.  .1.  ) )
)
4336, 40, 423eqtr4rd 2480 . . 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 10325 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  x.  y
)  e.  ZZ )
4544adantl 454 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( x  x.  y )  e.  ZZ )
46 oveq1 6089 . . . . 5  |-  ( n  =  ( x  x.  y )  ->  (
n  .x.  .1.  )  =  ( ( x  x.  y )  .x.  .1.  ) )
47 ovex 6107 . . . . 5  |-  ( ( x  x.  y ) 
.x.  .1.  )  e.  _V
4846, 18, 47fvmpt 5807 . . . 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 6089 . . . . . 6  |-  ( n  =  x  ->  (
n  .x.  .1.  )  =  ( x  .x.  .1.  ) )
51 ovex 6107 . . . . . 6  |-  ( x 
.x.  .1.  )  e.  _V
5250, 18, 51fvmpt 5807 . . . . 5  |-  ( x  e.  ZZ  ->  ( F `  x )  =  ( x  .x.  .1.  ) )
53 oveq1 6089 . . . . . 6  |-  ( n  =  y  ->  (
n  .x.  .1.  )  =  ( y  .x.  .1.  ) )
54 ovex 6107 . . . . . 6  |-  ( y 
.x.  .1.  )  e.  _V
5553, 18, 54fvmpt 5807 . . . . 5  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  .1.  ) )
5652, 55oveqan12d 6101 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( F `  x ) ( .r
`  R ) ( F `  y ) )  =  ( ( x  .x.  .1.  )
( .r `  R
) ( y  .x.  .1.  ) ) )
5756adantl 454 . . 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 2479 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  ZZ  /\  y  e.  ZZ )
)  ->  ( F `  ( x  x.  y
) )  =  ( ( F `  x
) ( .r `  R ) ( F `
 y ) ) )
592, 24, 18, 22mulgghm2 16787 . . 3  |-  ( ( R  e.  Grp  /\  .1.  e.  ( Base `  R
) )  ->  F  e.  ( Z  GrpHom  R ) )
6028, 23, 59syl2anc 644 . 2  |-  ( R  e.  Ring  ->  F  e.  ( Z  GrpHom  R ) )
614, 7, 8, 11, 12, 14, 15, 27, 58, 60isrhm2d 15830 1  |-  ( R  e.  Ring  ->  F  e.  ( Z RingHom  R )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4267   ` cfv 5455  (class class class)co 6082   1c1 8992    x. cmul 8996   ZZcz 10283   Basecbs 13470   ↾s cress 13471   .rcmulr 13531   Grpcgrp 14686  .gcmg 14690    GrpHom cghm 15004   Ringcrg 15661   1rcur 15663   RingHom crh 15818  SubRingcsubrg 15865  ℂfldccnfld 16704
This theorem is referenced by:  mulgrhm2  16789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-fz 11045  df-seq 11325  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-0g 13728  df-mnd 14691  df-mhm 14739  df-grp 14813  df-minusg 14814  df-mulg 14816  df-subg 14942  df-ghm 15005  df-cmn 15415  df-mgp 15650  df-rng 15664  df-cring 15665  df-ur 15666  df-rnghom 15820  df-subrg 15867  df-cnfld 16705
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