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Theorem proot1hash 30793
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 2467 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 proot1hash.o . . . . . 6  |-  O  =  ( od `  G
)
31, 2odf 16367 . . . . 5  |-  O :
( Base `  G ) --> NN0
4 ffn 5731 . . . . 5  |-  ( O : ( Base `  G
) --> NN0  ->  O  Fn  ( Base `  G )
)
5 fniniseg2 6004 . . . . 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 998 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N } ) )
8 fniniseg 6002 . . . . . . . . . 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 196 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( X  e.  ( Base `  G
)  /\  ( O `  X )  =  N ) )
1110simprd 463 . . . . . . 7  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  =  N )
1211eqeq2d 2481 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( ( O `
 x )  =  ( O `  X
)  <->  ( O `  x )  =  N ) )
1312rabbidv 3105 . . . . 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 17752 . . . . . . . . . 10  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
1514simprbi 464 . . . . . . . . 9  |-  ( R  e. IDomn  ->  R  e. Domn )
16153ad2ant1 1017 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  R  e. Domn )
17 domnrng 17744 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
18 eqid 2467 . . . . . . . . 9  |-  (Unit `  R )  =  (Unit `  R )
19 proot1hash.g . . . . . . . . 9  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
2018, 19unitgrp 17117 . . . . . . . 8  |-  ( R  e.  Ring  ->  G  e. 
Grp )
2116, 17, 203syl 20 . . . . . . 7  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  G  e.  Grp )
221subgacs 16041 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
23 acsmre 14907 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2421, 22, 233syl 20 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
25 eqid 2467 . . . . . . 7  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
2625mrcssv 14869 . . . . . 6  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  { X } )  C_  ( Base `  G ) )
27 dfrab3ss 3776 . . . . . 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 20 . . . . 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 3691 . . . . . 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 999 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  R  e. IDomn )
31 simpl2 1000 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  N  e.  NN )
32 simpr 461 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  x  e.  ( `' O " { N }
) )
33 simpl3 1001 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N }
) )
3419, 2, 25proot1mul 30789 . . . . . . . . . . 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 1229 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } ) )
3635ex 434 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( x  e.  ( `' O " { N } )  ->  x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } ) ) )
3736ssrdv 3510 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 { X }
) )
386, 37syl5eqssr 3549 . . . . . . 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 3490 . . . . . . 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 196 . . . . . 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 2520 . . . . 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 2513 . . . 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 2520 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } )  =  { x  e.  ( (mrCls `  (SubGrp `  G ) ) `  { X } )  |  ( O `  x
)  =  ( O `
 X ) } )
4443fveq2d 5870 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( # `  ( `' O " { N } ) )  =  ( # `  {
x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } )  |  ( O `  x )  =  ( O `  X ) } ) )
4510simpld 459 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  (
Base `  G )
)
46 simp2 997 . . . 4  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  N  e.  NN )
4711, 46eqeltrd 2555 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  e.  NN )
481, 2, 25odngen 16403 . . 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 1228 . 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 5870 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( phi `  ( O `  X ) )  =  ( phi `  N ) )
5144, 49, 503eqtrd 2512 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818    i^i cin 3475    C_ wss 3476   {csn 4027   `'ccnv 4998   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   NNcn 10536   NN0cn0 10795   #chash 12373   phicphi 14153   Basecbs 14490   ↾s cress 14491  Moorecmre 14837  mrClscmrc 14838  ACScacs 14840   Grpcgrp 15727  SubGrpcsubg 16000   odcod 16355  mulGrpcmgp 16943   Ringcrg 17000   CRingccrg 17001  Unitcui 17089  Domncdomn 17727  IDomncidom 17728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-ofr 6525  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-dvds 13848  df-gcd 14004  df-phi 14155  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-0g 14697  df-gsum 14698  df-prds 14703  df-pws 14705  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-eqg 16005  df-ghm 16070  df-cntz 16160  df-od 16359  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-srg 16960  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-rnghom 17165  df-subrg 17227  df-lmod 17314  df-lss 17379  df-lsp 17418  df-nzr 17705  df-rlreg 17730  df-domn 17731  df-idom 17732  df-assa 17760  df-asp 17761  df-ascl 17762  df-psr 17804  df-mvr 17805  df-mpl 17806  df-opsr 17808  df-evls 17970  df-evl 17971  df-psr1 18018  df-vr1 18019  df-ply1 18020  df-coe1 18021  df-evl1 18152  df-cnfld 18220  df-mdeg 22216  df-deg1 22217  df-mon1 22294  df-uc1p 22295  df-q1p 22296  df-r1p 22297
This theorem is referenced by: (None)
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