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Theorem proot1mul 27515
Description: Any primitive  N-th root of unity is a multiple of any other. (Contributed by Stefan O'Rear, 2-Nov-2015.)
Hypotheses
Ref Expression
idomsubgmo.g  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
proot1mul.o  |-  O  =  ( od `  G
)
proot1mul.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
Assertion
Ref Expression
proot1mul  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  X  e.  ( K `  { Y } ) )

Proof of Theorem proot1mul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 730 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  R  e. IDomn )
2 isidom 16045 . . . . . . . 8  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
32simprbi 450 . . . . . . 7  |-  ( R  e. IDomn  ->  R  e. Domn )
4 domnrng 16037 . . . . . . 7  |-  ( R  e. Domn  ->  R  e.  Ring )
53, 4syl 15 . . . . . 6  |-  ( R  e. IDomn  ->  R  e.  Ring )
6 eqid 2283 . . . . . . 7  |-  (Unit `  R )  =  (Unit `  R )
7 idomsubgmo.g . . . . . . 7  |-  G  =  ( (mulGrp `  R
)s  (Unit `  R )
)
86, 7unitgrp 15449 . . . . . 6  |-  ( R  e.  Ring  ->  G  e. 
Grp )
91, 5, 83syl 18 . . . . 5  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  G  e.  Grp )
10 eqid 2283 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1110subgacs 14652 . . . . 5  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
12 acsmre 13554 . . . . 5  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
139, 11, 123syl 18 . . . 4  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G ) ) )
14 simprl 732 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  X  e.  ( `' O " { N } ) )
15 proot1mul.o . . . . . . . . 9  |-  O  =  ( od `  G
)
1610, 15odf 14852 . . . . . . . 8  |-  O :
( Base `  G ) --> NN0
17 ffn 5389 . . . . . . . 8  |-  ( O : ( Base `  G
) --> NN0  ->  O  Fn  ( Base `  G )
)
18 fniniseg 5646 . . . . . . . 8  |-  ( O  Fn  ( Base `  G
)  ->  ( X  e.  ( `' O " { N } )  <->  ( X  e.  ( Base `  G
)  /\  ( O `  X )  =  N ) ) )
1916, 17, 18mp2b 9 . . . . . . 7  |-  ( X  e.  ( `' O " { N } )  <-> 
( X  e.  (
Base `  G )  /\  ( O `  X
)  =  N ) )
2014, 19sylib 188 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( X  e.  ( Base `  G
)  /\  ( O `  X )  =  N ) )
2120simpld 445 . . . . 5  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  X  e.  (
Base `  G )
)
2221snssd 3760 . . . 4  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  { X }  C_  ( Base `  G
) )
23 proot1mul.k . . . . 5  |-  K  =  (mrCls `  (SubGrp `  G
) )
2423mrcssid 13519 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  { X }  C_  ( Base `  G ) )  ->  { X }  C_  ( K `  { X } ) )
2513, 22, 24syl2anc 642 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  { X }  C_  ( K `  { X } ) )
26 snssg 3754 . . . 4  |-  ( X  e.  ( `' O " { N } )  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
2714, 26syl 15 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( X  e.  ( K `  { X } )  <->  { X }  C_  ( K `  { X } ) ) )
2825, 27mpbird 223 . 2  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  X  e.  ( K `  { X } ) )
297idomsubgmo 27514 . . . 4  |-  ( ( R  e. IDomn  /\  N  e.  NN )  ->  E* x  e.  (SubGrp `  G
) ( # `  x
)  =  N )
3029adantr 451 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  E* x  e.  (SubGrp `  G )
( # `  x )  =  N )
3123mrccl 13513 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  { X }  C_  ( Base `  G ) )  ->  ( K `  { X } )  e.  (SubGrp `  G )
)
3213, 22, 31syl2anc 642 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( K `  { X } )  e.  (SubGrp `  G )
)
3320simprd 449 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( O `  X )  =  N )
34 simplr 731 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  N  e.  NN )
3533, 34eqeltrd 2357 . . . . 5  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( O `  X )  e.  NN )
3610, 15, 23odhash2 14886 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  ( Base `  G )  /\  ( O `  X )  e.  NN )  ->  ( # `
 ( K `  { X } ) )  =  ( O `  X ) )
379, 21, 35, 36syl3anc 1182 . . . 4  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( # `  ( K `  { X } ) )  =  ( O `  X
) )
3837, 33eqtrd 2315 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( # `  ( K `  { X } ) )  =  N )
39 simprr 733 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  Y  e.  ( `' O " { N } ) )
40 fniniseg 5646 . . . . . . . 8  |-  ( O  Fn  ( Base `  G
)  ->  ( Y  e.  ( `' O " { N } )  <->  ( Y  e.  ( Base `  G
)  /\  ( O `  Y )  =  N ) ) )
4116, 17, 40mp2b 9 . . . . . . 7  |-  ( Y  e.  ( `' O " { N } )  <-> 
( Y  e.  (
Base `  G )  /\  ( O `  Y
)  =  N ) )
4239, 41sylib 188 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( Y  e.  ( Base `  G
)  /\  ( O `  Y )  =  N ) )
4342simpld 445 . . . . 5  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  Y  e.  (
Base `  G )
)
4443snssd 3760 . . . 4  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  { Y }  C_  ( Base `  G
) )
4523mrccl 13513 . . . 4  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  { Y }  C_  ( Base `  G ) )  ->  ( K `  { Y } )  e.  (SubGrp `  G )
)
4613, 44, 45syl2anc 642 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( K `  { Y } )  e.  (SubGrp `  G )
)
4742simprd 449 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( O `  Y )  =  N )
4847, 34eqeltrd 2357 . . . . 5  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( O `  Y )  e.  NN )
4910, 15, 23odhash2 14886 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  ( Base `  G )  /\  ( O `  Y )  e.  NN )  ->  ( # `
 ( K `  { Y } ) )  =  ( O `  Y ) )
509, 43, 48, 49syl3anc 1182 . . . 4  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( # `  ( K `  { Y } ) )  =  ( O `  Y
) )
5150, 47eqtrd 2315 . . 3  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( # `  ( K `  { Y } ) )  =  N )
52 fveq2 5525 . . . . 5  |-  ( x  =  ( K `  { X } )  -> 
( # `  x )  =  ( # `  ( K `  { X } ) ) )
5352eqeq1d 2291 . . . 4  |-  ( x  =  ( K `  { X } )  -> 
( ( # `  x
)  =  N  <->  ( # `  ( K `  { X } ) )  =  N ) )
54 fveq2 5525 . . . . 5  |-  ( x  =  ( K `  { Y } )  -> 
( # `  x )  =  ( # `  ( K `  { Y } ) ) )
5554eqeq1d 2291 . . . 4  |-  ( x  =  ( K `  { Y } )  -> 
( ( # `  x
)  =  N  <->  ( # `  ( K `  { Y } ) )  =  N ) )
5653, 55rmoi 3080 . . 3  |-  ( ( E* x  e.  (SubGrp `  G ) ( # `  x )  =  N  /\  ( ( K `
 { X }
)  e.  (SubGrp `  G )  /\  ( # `
 ( K `  { X } ) )  =  N )  /\  ( ( K `  { Y } )  e.  (SubGrp `  G )  /\  ( # `  ( K `  { Y } ) )  =  N ) )  -> 
( K `  { X } )  =  ( K `  { Y } ) )
5730, 32, 38, 46, 51, 56syl122anc 1191 . 2  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  ( K `  { X } )  =  ( K `  { Y } ) )
5828, 57eleqtrd 2359 1  |-  ( ( ( R  e. IDomn  /\  N  e.  NN )  /\  ( X  e.  ( `' O " { N }
)  /\  Y  e.  ( `' O " { N } ) ) )  ->  X  e.  ( K `  { Y } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E*wrmo 2546    C_ wss 3152   {csn 3640   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   NNcn 9746   NN0cn0 9965   #chash 11337   Basecbs 13148   ↾s cress 13149  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Grpcgrp 14362  SubGrpcsubg 14615   odcod 14840  mulGrpcmgp 15325   Ringcrg 15337   CRingccrg 15338  Unitcui 15421  Domncdomn 16021  IDomncidom 16022
This theorem is referenced by:  proot1hash  27519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-od 14844  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-nzr 16010  df-rlreg 16024  df-domn 16025  df-idom 16026  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101  df-evl 16102  df-opsr 16106  df-psr1 16257  df-vr1 16258  df-ply1 16259  df-evl1 16261  df-coe1 16262  df-cnfld 16378  df-mdeg 19441  df-deg1 19442  df-mon1 19516  df-uc1p 19517  df-q1p 19518  df-r1p 19519
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