Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  proot1hash Structured version   Unicode version

Theorem proot1hash 27498
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 2438 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 proot1hash.o . . . . . 6  |-  O  =  ( od `  G
)
31, 2odf 15177 . . . . 5  |-  O :
( Base `  G ) --> NN0
4 ffn 5593 . . . . 5  |-  ( O : ( Base `  G
) --> NN0  ->  O  Fn  ( Base `  G )
)
5 fniniseg2 5855 . . . . 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 960 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N } ) )
8 fniniseg 5853 . . . . . . . . . 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 190 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( X  e.  ( Base `  G
)  /\  ( O `  X )  =  N ) )
1110simprd 451 . . . . . . 7  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  =  N )
1211eqeq2d 2449 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( ( O `
 x )  =  ( O `  X
)  <->  ( O `  x )  =  N ) )
1312rabbidv 2950 . . . . 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 16366 . . . . . . . . . 10  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
1514simprbi 452 . . . . . . . . 9  |-  ( R  e. IDomn  ->  R  e. Domn )
16153ad2ant1 979 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  R  e. Domn )
17 domnrng 16358 . . . . . . . 8  |-  ( R  e. Domn  ->  R  e.  Ring )
18 eqid 2438 . . . . . . . . 9  |-  (Unit `  R )  =  (Unit `  R )
19 proot1hash.g . . . . . . . . 9  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
2018, 19unitgrp 15774 . . . . . . . 8  |-  ( R  e.  Ring  ->  G  e. 
Grp )
2116, 17, 203syl 19 . . . . . . 7  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  G  e.  Grp )
221subgacs 14977 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
23 acsmre 13879 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2421, 22, 233syl 19 . . . . . 6  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
25 eqid 2438 . . . . . . 7  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
2625mrcssv 13841 . . . . . 6  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  { X } )  C_  ( Base `  G ) )
27 dfrab3ss 3621 . . . . . 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 19 . . . . 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 3535 . . . . . 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 961 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  R  e. IDomn )
31 simpl2 962 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  N  e.  NN )
32 simpr 449 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  x  e.  ( `' O " { N }
) )
33 simpl3 963 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  X  e.  ( `' O " { N }
) )
3419, 2, 25proot1mul 27494 . . . . . . . . . . 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 1186 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  /\  x  e.  ( `' O " { N } ) )  ->  x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } ) )
3635ex 425 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( x  e.  ( `' O " { N } )  ->  x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } ) ) )
3736ssrdv 3356 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( `' O " { N } ) 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 { X }
) )
386, 37syl5eqssr 3395 . . . . . . 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 3336 . . . . . . 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 190 . . . . . 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 5734 . 2  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( # `  ( `' O " { N } ) )  =  ( # `  {
x  e.  ( (mrCls `  (SubGrp `  G )
) `  { X } )  |  ( O `  x )  =  ( O `  X ) } ) )
4510simpld 447 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  X  e.  (
Base `  G )
)
46 simp2 959 . . . 4  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  N  e.  NN )
4711, 46eqeltrd 2512 . . 3  |-  ( ( R  e. IDomn  /\  N  e.  NN  /\  X  e.  ( `' O " { N } ) )  ->  ( O `  X )  e.  NN )
481, 2, 25odngen 15213 . . 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 1185 . 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 5734 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711    i^i cin 3321    C_ wss 3322   {csn 3816   `'ccnv 4879   "cima 4883    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   NNcn 10002   NN0cn0 10223   #chash 11620   phicphi 13155   Basecbs 13471   ↾s cress 13472  Moorecmre 13809  mrClscmrc 13810  ACScacs 13812   Grpcgrp 14687  SubGrpcsubg 14940   odcod 15165  mulGrpcmgp 15650   Ringcrg 15662   CRingccrg 15663  Unitcui 15746  Domncdomn 16342  IDomncidom 16343
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 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-ofr 6308  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-acn 7831  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-dvds 12855  df-gcd 13009  df-phi 13157  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-prds 13673  df-pws 13675  df-0g 13729  df-gsum 13730  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-subg 14943  df-eqg 14945  df-ghm 15006  df-cntz 15118  df-od 15169  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-rnghom 15821  df-subrg 15868  df-lmod 15954  df-lss 16011  df-lsp 16050  df-nzr 16331  df-rlreg 16345  df-domn 16346  df-idom 16347  df-assa 16374  df-asp 16375  df-ascl 16376  df-psr 16419  df-mvr 16420  df-mpl 16421  df-evls 16422  df-evl 16423  df-opsr 16427  df-psr1 16578  df-vr1 16579  df-ply1 16580  df-evl1 16582  df-coe1 16583  df-cnfld 16706  df-mdeg 19980  df-deg1 19981  df-mon1 20055  df-uc1p 20056  df-q1p 20057  df-r1p 20058
  Copyright terms: Public domain W3C validator