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Theorem proot1hash 27622
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 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 proot1hash.o . . . . . 6  |-  O  =  ( od `  G
)
31, 2odf 14868 . . . . 5  |-  O :
( Base `  G ) --> NN0
4 ffn 5405 . . . . 5  |-  ( O : ( Base `  G
) --> NN0  ->  O  Fn  ( Base `  G )
)
5 fniniseg2 5664 . . . . 5  |-  ( O  Fn  ( Base `  G
)  ->  ( `' O " { N }
)  =  { x  e.  ( Base `  G
)  |  ( O `
 x )  =  N } )
63, 4, 5mp2b 9 . . . 4  |-  ( `' O " { N } )  =  {
x  e.  ( Base `  G )  |  ( O `  x )  =  N }
7 simp3 957 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N } ) )
8 fniniseg 5662 . . . . . . . . . 10  |-  ( O  Fn  ( Base `  G
)  ->  ( X  e.  ( `' O " { N } )  <->  ( X  e.  ( Base `  G
)  /\  ( O `  X )  =  N ) ) )
93, 4, 8mp2b 9 . . . . . . . . 9  |-  ( X  e.  ( `' O " { N } )  <-> 
( X  e.  (
Base `  G )  /\  ( O `  X
)  =  N ) )
107, 9sylib 188 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( X  e.  ( Base `  G
)  /\  ( O `  X )  =  N ) )
1110simprd 449 . . . . . . 7  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  =  N )
1211eqeq2d 2307 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( ( O `
 x )  =  ( O `  X
)  <->  ( O `  x )  =  N ) )
1312rabbidv 2793 . . . . 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 16061 . . . . . . . . . 10  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
1514simprbi 450 . . . . . . . . 9  |-  ( R  e. IDomn  ->  R  e. Domn )
16153ad2ant1 976 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  R  e. Domn )
17 domnrng 16053 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
18 eqid 2296 . . . . . . . . 9  |-  (Unit `  R )  =  (Unit `  R )
19 proot1hash.g . . . . . . . . 9  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
2018, 19unitgrp 15465 . . . . . . . 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 14668 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
23 acsmre 13570 . . . . . . 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 2296 . . . . . . 7  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
2625mrcssv 13532 . . . . . 6  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  { X } )  C_  ( Base `  G ) )
27 dfrab3ss 3459 . . . . . 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 3374 . . . . . 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 958 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  R  e. IDomn )
31 simpl2 959 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  N  e.  NN )
32 simpr 447 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  x  e.  ( `' O " { N }
) )
33 simpl3 960 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N }
) )
3419, 2, 25proot1mul 27618 . . . . . . . . . . 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 1183 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } ) )
3635ex 423 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( x  e.  ( `' O " { N } )  ->  x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } ) ) )
3736ssrdv 3198 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 { X }
) )
386, 37syl5eqssr 3236 . . . . . . 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 3179 . . . . . . 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 188 . . . . . 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 2340 . . . . 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 2333 . . . 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 2340 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } )  =  { x  e.  ( (mrCls `  (SubGrp `  G ) ) `  { X } )  |  ( O `  x
)  =  ( O `
 X ) } )
4443fveq2d 5545 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( # `  ( `' O " { N } ) )  =  ( # `  {
x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } )  |  ( O `  x )  =  ( O `  X ) } ) )
4510simpld 445 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  (
Base `  G )
)
46 simp2 956 . . . 4  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  N  e.  NN )
4711, 46eqeltrd 2370 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  e.  NN )
481, 2, 25odngen 14904 . . 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 1182 . 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 5545 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( phi `  ( O `  X ) )  =  ( phi `  N ) )
5144, 49, 503eqtrd 2332 1  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( # `  ( `' O " { N } ) )  =  ( phi `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560    i^i cin 3164    C_ wss 3165   {csn 3653   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   NNcn 9762   NN0cn0 9981   #chash 11353   phicphi 12848   Basecbs 13164   ↾s cress 13165  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378  SubGrpcsubg 14631   odcod 14856  mulGrpcmgp 15341   Ringcrg 15353   CRingccrg 15354  Unitcui 15437  Domncdomn 16037  IDomncidom 16038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-phi 12850  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-eqg 14636  df-ghm 14697  df-cntz 14809  df-od 14860  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-nzr 16026  df-rlreg 16040  df-domn 16041  df-idom 16042  df-assa 16069  df-asp 16070  df-ascl 16071  df-psr 16114  df-mvr 16115  df-mpl 16116  df-evls 16117  df-evl 16118  df-opsr 16122  df-psr1 16273  df-vr1 16274  df-ply1 16275  df-evl1 16277  df-coe1 16278  df-cnfld 16394  df-mdeg 19457  df-deg1 19458  df-mon1 19532  df-uc1p 19533  df-q1p 19534  df-r1p 19535
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