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Theorem proot1hash 35791
Description: If an integral domain has a primitive  N-th root of unity, it has exactly  ( phi `  N ) of them. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Hypotheses
Ref Expression
proot1hash.g  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
proot1hash.o  |-  O  =  ( od `  G
)
Assertion
Ref Expression
proot1hash  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( # `  ( `' O " { N } ) )  =  ( phi `  N
) )

Proof of Theorem proot1hash
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2429 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 proot1hash.o . . . . . 6  |-  O  =  ( od `  G
)
31, 2odf 17132 . . . . 5  |-  O :
( Base `  G ) --> NN0
4 ffn 5746 . . . . 5  |-  ( O : ( Base `  G
) --> NN0  ->  O  Fn  ( Base `  G )
)
5 fniniseg2 6020 . . . . 5  |-  ( O  Fn  ( Base `  G
)  ->  ( `' O " { N }
)  =  { x  e.  ( Base `  G
)  |  ( O `
 x )  =  N } )
63, 4, 5mp2b 10 . . . 4  |-  ( `' O " { N } )  =  {
x  e.  ( Base `  G )  |  ( O `  x )  =  N }
7 simp3 1007 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N } ) )
8 fniniseg 6018 . . . . . . . . . 10  |-  ( O  Fn  ( Base `  G
)  ->  ( X  e.  ( `' O " { N } )  <->  ( X  e.  ( Base `  G
)  /\  ( O `  X )  =  N ) ) )
93, 4, 8mp2b 10 . . . . . . . . 9  |-  ( X  e.  ( `' O " { N } )  <-> 
( X  e.  (
Base `  G )  /\  ( O `  X
)  =  N ) )
107, 9sylib 199 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( X  e.  ( Base `  G
)  /\  ( O `  X )  =  N ) )
1110simprd 464 . . . . . . 7  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  =  N )
1211eqeq2d 2443 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( ( O `
 x )  =  ( O `  X
)  <->  ( O `  x )  =  N ) )
1312rabbidv 3079 . . . . 5  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  { x  e.  ( (mrCls `  (SubGrp `  G ) ) `  { X } )  |  ( O `  x
)  =  ( O `
 X ) }  =  { x  e.  ( (mrCls `  (SubGrp `  G ) ) `  { X } )  |  ( O `  x
)  =  N }
)
14 isidom 18467 . . . . . . . . . 10  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
1514simprbi 465 . . . . . . . . 9  |-  ( R  e. IDomn  ->  R  e. Domn )
16153ad2ant1 1026 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  R  e. Domn )
17 domnring 18459 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
18 eqid 2429 . . . . . . . . 9  |-  (Unit `  R )  =  (Unit `  R )
19 proot1hash.g . . . . . . . . 9  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
2018, 19unitgrp 17834 . . . . . . . 8  |-  ( R  e.  Ring  ->  G  e. 
Grp )
2116, 17, 203syl 18 . . . . . . 7  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  G  e.  Grp )
221subgacs 16807 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
23 acsmre 15513 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2421, 22, 233syl 18 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
25 eqid 2429 . . . . . . 7  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
2625mrcssv 15475 . . . . . 6  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  { X } )  C_  ( Base `  G ) )
27 dfrab3ss 3757 . . . . . 6  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  { X } )  C_  ( Base `  G )  ->  { x  e.  (
(mrCls `  (SubGrp `  G
) ) `  { X } )  |  ( O `  x )  =  N }  =  ( ( (mrCls `  (SubGrp `  G ) ) `
 { X }
)  i^i  { x  e.  ( Base `  G
)  |  ( O `
 x )  =  N } ) )
2824, 26, 273syl 18 . . . . 5  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  { x  e.  ( (mrCls `  (SubGrp `  G ) ) `  { X } )  |  ( O `  x
)  =  N }  =  ( ( (mrCls `  (SubGrp `  G )
) `  { X } )  i^i  {
x  e.  ( Base `  G )  |  ( O `  x )  =  N } ) )
29 incom 3661 . . . . . 6  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  { X } )  i^i  {
x  e.  ( Base `  G )  |  ( O `  x )  =  N } )  =  ( { x  e.  ( Base `  G
)  |  ( O `
 x )  =  N }  i^i  (
(mrCls `  (SubGrp `  G
) ) `  { X } ) )
30 simpl1 1008 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  R  e. IDomn )
31 simpl2 1009 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  N  e.  NN )
32 simpr 462 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  x  e.  ( `' O " { N }
) )
33 simpl3 1010 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N }
) )
3419, 2, 25proot1mul 35787 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  (
x  e.  ( `' O " { N } )  /\  X  e.  ( `' O " { N } ) ) )  ->  x  e.  ( (mrCls `  (SubGrp `  G
) ) `  { X } ) )
3530, 31, 32, 33, 34syl22anc 1265 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } ) )
3635ex 435 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( x  e.  ( `' O " { N } )  ->  x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } ) ) )
3736ssrdv 3476 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 { X }
) )
386, 37syl5eqssr 3515 . . . . . . 7  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  { x  e.  ( Base `  G
)  |  ( O `
 x )  =  N }  C_  (
(mrCls `  (SubGrp `  G
) ) `  { X } ) )
39 df-ss 3456 . . . . . . 7  |-  ( { x  e.  ( Base `  G )  |  ( O `  x )  =  N }  C_  ( (mrCls `  (SubGrp `  G
) ) `  { X } )  <->  ( {
x  e.  ( Base `  G )  |  ( O `  x )  =  N }  i^i  ( (mrCls `  (SubGrp `  G
) ) `  { X } ) )  =  { x  e.  (
Base `  G )  |  ( O `  x )  =  N } )
4038, 39sylib 199 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( { x  e.  ( Base `  G
)  |  ( O `
 x )  =  N }  i^i  (
(mrCls `  (SubGrp `  G
) ) `  { X } ) )  =  { x  e.  (
Base `  G )  |  ( O `  x )  =  N } )
4129, 40syl5eq 2482 . . . . 5  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( ( (mrCls `  (SubGrp `  G )
) `  { X } )  i^i  {
x  e.  ( Base `  G )  |  ( O `  x )  =  N } )  =  { x  e.  ( Base `  G
)  |  ( O `
 x )  =  N } )
4213, 28, 413eqtrrd 2475 . . . 4  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  { x  e.  ( Base `  G
)  |  ( O `
 x )  =  N }  =  {
x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } )  |  ( O `  x )  =  ( O `  X ) } )
436, 42syl5eq 2482 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } )  =  { x  e.  ( (mrCls `  (SubGrp `  G ) ) `  { X } )  |  ( O `  x
)  =  ( O `
 X ) } )
4443fveq2d 5885 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( # `  ( `' O " { N } ) )  =  ( # `  {
x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } )  |  ( O `  x )  =  ( O `  X ) } ) )
4510simpld 460 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  (
Base `  G )
)
46 simp2 1006 . . . 4  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  N  e.  NN )
4711, 46eqeltrd 2517 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  e.  NN )
481, 2, 25odngen 17168 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  ( Base `  G )  /\  ( O `  X )  e.  NN )  ->  ( # `
 { x  e.  ( (mrCls `  (SubGrp `  G ) ) `  { X } )  |  ( O `  x
)  =  ( O `
 X ) } )  =  ( phi `  ( O `  X
) ) )
4921, 45, 47, 48syl3anc 1264 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( # `  {
x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } )  |  ( O `  x )  =  ( O `  X ) } )  =  ( phi `  ( O `  X ) ) )
5011fveq2d 5885 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( phi `  ( O `  X ) )  =  ( phi `  N ) )
5144, 49, 503eqtrd 2474 1  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( # `  ( `' O " { N } ) )  =  ( phi `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {crab 2786    i^i cin 3441    C_ wss 3442   {csn 4002   `'ccnv 4853   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   NNcn 10609   NN0cn0 10869   #chash 12512   phicphi 14681   Basecbs 15084   ↾s cress 15085  Moorecmre 15443  mrClscmrc 15444  ACScacs 15446   Grpcgrp 16624  SubGrpcsubg 16766   odcod 17120  mulGrpcmgp 17662   Ringcrg 17719   CRingccrg 17720  Unitcui 17806  Domncdomn 18443  IDomncidom 18444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-disj 4398  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-ofr 6546  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-omul 7195  df-er 7371  df-ec 7373  df-qs 7377  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-acn 8375  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-dvds 14284  df-gcd 14443  df-phi 14683  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15166  df-mulr 15167  df-starv 15168  df-sca 15169  df-vsca 15170  df-ip 15171  df-tset 15172  df-ple 15173  df-ds 15175  df-unif 15176  df-hom 15177  df-cco 15178  df-0g 15303  df-gsum 15304  df-prds 15309  df-pws 15311  df-mre 15447  df-mrc 15448  df-acs 15450  df-mgm 16443  df-sgrp 16482  df-mnd 16492  df-mhm 16537  df-submnd 16538  df-grp 16628  df-minusg 16629  df-sbg 16630  df-mulg 16631  df-subg 16769  df-eqg 16771  df-ghm 16836  df-cntz 16926  df-od 17124  df-cmn 17371  df-abl 17372  df-mgp 17663  df-ur 17675  df-srg 17679  df-ring 17721  df-cring 17722  df-oppr 17790  df-dvdsr 17808  df-unit 17809  df-invr 17839  df-rnghom 17882  df-subrg 17945  df-lmod 18032  df-lss 18095  df-lsp 18134  df-nzr 18421  df-rlreg 18446  df-domn 18447  df-idom 18448  df-assa 18475  df-asp 18476  df-ascl 18477  df-psr 18519  df-mvr 18520  df-mpl 18521  df-opsr 18523  df-evls 18668  df-evl 18669  df-psr1 18712  df-vr1 18713  df-ply1 18714  df-coe1 18715  df-evl1 18844  df-cnfld 18910  df-mdeg 22889  df-deg1 22890  df-mon1 22964  df-uc1p 22965  df-q1p 22966  df-r1p 22967
This theorem is referenced by: (None)
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