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Theorem proot1hash 29565
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 2441 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 proot1hash.o . . . . . 6  |-  O  =  ( od `  G
)
31, 2odf 16038 . . . . 5  |-  O :
( Base `  G ) --> NN0
4 ffn 5557 . . . . 5  |-  ( O : ( Base `  G
) --> NN0  ->  O  Fn  ( Base `  G )
)
5 fniniseg2 5824 . . . . 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 990 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N } ) )
8 fniniseg 5822 . . . . . . . . . 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 2452 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( ( O `
 x )  =  ( O `  X
)  <->  ( O `  x )  =  N ) )
1312rabbidv 2962 . . . . 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 17374 . . . . . . . . . 10  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
1514simprbi 464 . . . . . . . . 9  |-  ( R  e. IDomn  ->  R  e. Domn )
16153ad2ant1 1009 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  R  e. Domn )
17 domnrng 17366 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
18 eqid 2441 . . . . . . . . 9  |-  (Unit `  R )  =  (Unit `  R )
19 proot1hash.g . . . . . . . . 9  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
2018, 19unitgrp 16757 . . . . . . . 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 15714 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
23 acsmre 14588 . . . . . . 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 2441 . . . . . . 7  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
2625mrcssv 14550 . . . . . 6  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  { X } )  C_  ( Base `  G ) )
27 dfrab3ss 3626 . . . . . 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 3541 . . . . . 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 991 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  R  e. IDomn )
31 simpl2 992 . . . . . . . . . . 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 993 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N }
) )
3419, 2, 25proot1mul 29561 . . . . . . . . . . 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 1219 . . . . . . . . . 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 3360 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 { X }
) )
386, 37syl5eqssr 3399 . . . . . . 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 3340 . . . . . . 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 2485 . . . . 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 2478 . . . 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 2485 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } )  =  { x  e.  ( (mrCls `  (SubGrp `  G ) ) `  { X } )  |  ( O `  x
)  =  ( O `
 X ) } )
4443fveq2d 5693 . 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 989 . . . 4  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  N  e.  NN )
4711, 46eqeltrd 2515 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  e.  NN )
481, 2, 25odngen 16074 . . 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 1218 . 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 5693 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( phi `  ( O `  X ) )  =  ( phi `  N ) )
5144, 49, 503eqtrd 2477 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 965    = wceq 1369    e. wcel 1756   {crab 2717    i^i cin 3325    C_ wss 3326   {csn 3875   `'ccnv 4837   "cima 4841    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089   NNcn 10320   NN0cn0 10577   #chash 12101   phicphi 13837   Basecbs 14172   ↾s cress 14173  Moorecmre 14518  mrClscmrc 14519  ACScacs 14521   Grpcgrp 15408  SubGrpcsubg 15673   odcod 16026  mulGrpcmgp 16589   Ringcrg 16643   CRingccrg 16644  Unitcui 16729  Domncdomn 17349  IDomncidom 17350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-disj 4261  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-ofr 6319  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-tpos 6743  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-omul 6923  df-er 7099  df-ec 7101  df-qs 7105  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-sup 7689  df-oi 7722  df-card 8107  df-acn 8110  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-rp 10990  df-fz 11436  df-fzo 11547  df-fl 11640  df-mod 11707  df-seq 11805  df-exp 11864  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-sum 13162  df-dvds 13534  df-gcd 13689  df-phi 13839  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-0g 14378  df-gsum 14379  df-prds 14384  df-pws 14386  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-mhm 15462  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546  df-subg 15676  df-eqg 15678  df-ghm 15743  df-cntz 15833  df-od 16030  df-cmn 16277  df-abl 16278  df-mgp 16590  df-ur 16602  df-srg 16606  df-rng 16645  df-cring 16646  df-oppr 16713  df-dvdsr 16731  df-unit 16732  df-invr 16762  df-rnghom 16804  df-subrg 16861  df-lmod 16948  df-lss 17012  df-lsp 17051  df-nzr 17338  df-rlreg 17352  df-domn 17353  df-idom 17354  df-assa 17382  df-asp 17383  df-ascl 17384  df-psr 17421  df-mvr 17422  df-mpl 17423  df-opsr 17425  df-evls 17586  df-evl 17587  df-psr1 17634  df-vr1 17635  df-ply1 17636  df-coe1 17637  df-evl1 17749  df-cnfld 17817  df-mdeg 21522  df-deg1 21523  df-mon1 21600  df-uc1p 21601  df-q1p 21602  df-r1p 21603
This theorem is referenced by: (None)
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