MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablfacrplem Structured version   Unicode version

Theorem ablfacrplem 16899
Description: Lemma for ablfacrp2 16901. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
ablfacrp.b  |-  B  =  ( Base `  G
)
ablfacrp.o  |-  O  =  ( od `  G
)
ablfacrp.k  |-  K  =  { x  e.  B  |  ( O `  x )  ||  M }
ablfacrp.l  |-  L  =  { x  e.  B  |  ( O `  x )  ||  N }
ablfacrp.g  |-  ( ph  ->  G  e.  Abel )
ablfacrp.m  |-  ( ph  ->  M  e.  NN )
ablfacrp.n  |-  ( ph  ->  N  e.  NN )
ablfacrp.1  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
ablfacrp.2  |-  ( ph  ->  ( # `  B
)  =  ( M  x.  N ) )
Assertion
Ref Expression
ablfacrplem  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
Distinct variable groups:    x, B    x, G    x, O    x, M    x, N    ph, x
Allowed substitution hints:    K( x)    L( x)

Proof of Theorem ablfacrplem
Dummy variables  g  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nprmdvds1 14100 . . . . . . 7  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
21adantl 466 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  1 )
3 ablfacrp.1 . . . . . . . 8  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
43adantr 465 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( M  gcd  N )  =  1 )
54breq2d 4452 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( M  gcd  N
)  <->  p  ||  1 ) )
62, 5mtbird 301 . . . . 5  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  ( M  gcd  N
) )
7 ablfacrp.k . . . . . . . . . . . . . 14  |-  K  =  { x  e.  B  |  ( O `  x )  ||  M }
8 ablfacrp.g . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  e.  Abel )
9 ablfacrp.m . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M  e.  NN )
109nnzd 10954 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ZZ )
11 ablfacrp.o . . . . . . . . . . . . . . . 16  |-  O  =  ( od `  G
)
12 ablfacrp.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  G
)
1311, 12oddvdssubg 16647 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G ) )
148, 10, 13syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G
) )
157, 14syl5eqel 2552 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
1615ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  e.  (SubGrp `  G )
)
17 eqid 2460 . . . . . . . . . . . . 13  |-  ( Gs  K )  =  ( Gs  K )
1817subggrp 15992 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  ( Gs  K
)  e.  Grp )
1916, 18syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( Gs  K )  e.  Grp )
2017subgbas 15993 . . . . . . . . . . . . 13  |-  ( K  e.  (SubGrp `  G
)  ->  K  =  ( Base `  ( Gs  K
) ) )
2116, 20syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  =  ( Base `  ( Gs  K ) ) )
22 ablfacrp.2 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( # `  B
)  =  ( M  x.  N ) )
239nnnn0d 10841 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  M  e.  NN0 )
24 ablfacrp.n . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  NN )
2524nnnn0d 10841 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  NN0 )
2623, 25nn0mulcld 10846 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
2722, 26eqeltrd 2548 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
28 fvex 5867 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
2912, 28eqeltri 2544 . . . . . . . . . . . . . . . 16  |-  B  e. 
_V
30 hashclb 12385 . . . . . . . . . . . . . . . 16  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
3129, 30ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
3227, 31sylibr 212 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  Fin )
33 ssrab2 3578 . . . . . . . . . . . . . . 15  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
347, 33eqsstri 3527 . . . . . . . . . . . . . 14  |-  K  C_  B
35 ssfi 7730 . . . . . . . . . . . . . 14  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
3632, 34, 35sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Fin )
3736ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  e.  Fin )
3821, 37eqeltrrd 2549 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( Base `  ( Gs  K ) )  e.  Fin )
39 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  e.  Prime )
40 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  ( # `  K
) )
4121fveq2d 5861 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( # `
 K )  =  ( # `  ( Base `  ( Gs  K ) ) ) )
4240, 41breqtrd 4464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  ( # `  ( Base `  ( Gs  K ) ) ) )
43 eqid 2460 . . . . . . . . . . . 12  |-  ( Base `  ( Gs  K ) )  =  ( Base `  ( Gs  K ) )
44 eqid 2460 . . . . . . . . . . . 12  |-  ( od
`  ( Gs  K ) )  =  ( od
`  ( Gs  K ) )
4543, 44odcau 16413 . . . . . . . . . . 11  |-  ( ( ( ( Gs  K )  e.  Grp  /\  ( Base `  ( Gs  K ) )  e.  Fin  /\  p  e.  Prime )  /\  p  ||  ( # `  ( Base `  ( Gs  K ) ) ) )  ->  E. g  e.  ( Base `  ( Gs  K ) ) ( ( od
`  ( Gs  K ) ) `  g )  =  p )
4619, 38, 39, 42, 45syl31anc 1226 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  E. g  e.  ( Base `  ( Gs  K ) ) ( ( od `  ( Gs  K ) ) `  g )  =  p )
4721rexeqdv 3058 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p  <->  E. g  e.  ( Base `  ( Gs  K ) ) ( ( od
`  ( Gs  K ) ) `  g )  =  p ) )
4846, 47mpbird 232 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p )
4917, 11, 44subgod 16379 . . . . . . . . . . . . 13  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  K )  ->  ( O `  g )  =  ( ( od
`  ( Gs  K ) ) `  g ) )
5016, 49sylan 471 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  ( O `  g )  =  ( ( od
`  ( Gs  K ) ) `  g ) )
51 fveq2 5857 . . . . . . . . . . . . . . . 16  |-  ( x  =  g  ->  ( O `  x )  =  ( O `  g ) )
5251breq1d 4450 . . . . . . . . . . . . . . 15  |-  ( x  =  g  ->  (
( O `  x
)  ||  M  <->  ( O `  g )  ||  M
) )
5352, 7elrab2 3256 . . . . . . . . . . . . . 14  |-  ( g  e.  K  <->  ( g  e.  B  /\  ( O `  g )  ||  M ) )
5453simprbi 464 . . . . . . . . . . . . 13  |-  ( g  e.  K  ->  ( O `  g )  ||  M )
5554adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  ( O `  g )  ||  M )
5650, 55eqbrtrrd 4462 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  (
( od `  ( Gs  K ) ) `  g )  ||  M
)
57 breq1 4443 . . . . . . . . . . 11  |-  ( ( ( od `  ( Gs  K ) ) `  g )  =  p  ->  ( ( ( od `  ( Gs  K ) ) `  g
)  ||  M  <->  p  ||  M
) )
5856, 57syl5ibcom 220 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  (
( ( od `  ( Gs  K ) ) `  g )  =  p  ->  p  ||  M
) )
5958rexlimdva 2948 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p  ->  p  ||  M
) )
6048, 59mpd 15 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  M )
6160ex 434 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( # `  K
)  ->  p  ||  M
) )
6261anim1d 564 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  ( # `  K
)  /\  p  ||  N
)  ->  ( p  ||  M  /\  p  ||  N ) ) )
63 prmz 14069 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  ZZ )
6463adantl 466 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  p  e.  ZZ )
65 hashcl 12383 . . . . . . . . . 10  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
6636, 65syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
6766nn0zd 10953 . . . . . . . 8  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
6867adantr 465 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( # `  K
)  e.  ZZ )
6924nnzd 10954 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
7069adantr 465 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  N  e.  ZZ )
71 dvdsgcdb 14030 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( # `  K )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( p  ||  ( # `
 K )  /\  p  ||  N )  <->  p  ||  (
( # `  K )  gcd  N ) ) )
7264, 68, 70, 71syl3anc 1223 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  ( # `  K
)  /\  p  ||  N
)  <->  p  ||  ( (
# `  K )  gcd  N ) ) )
7310adantr 465 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  M  e.  ZZ )
74 dvdsgcdb 14030 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( p  ||  M  /\  p  ||  N )  <-> 
p  ||  ( M  gcd  N ) ) )
7564, 73, 70, 74syl3anc 1223 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  M  /\  p  ||  N )  <->  p  ||  ( M  gcd  N ) ) )
7662, 72, 753imtr3d 267 . . . . 5  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( ( # `  K
)  gcd  N )  ->  p  ||  ( M  gcd  N ) ) )
776, 76mtod 177 . . . 4  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  ( ( # `  K
)  gcd  N )
)
7877nrexdv 2913 . . 3  |-  ( ph  ->  -.  E. p  e. 
Prime  p  ||  ( (
# `  K )  gcd  N ) )
79 exprmfct 14099 . . 3  |-  ( ( ( # `  K
)  gcd  N )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( ( # `  K )  gcd  N
) )
8078, 79nsyl 121 . 2  |-  ( ph  ->  -.  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) )
8124nnne0d 10569 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
82 simpr 461 . . . . . . 7  |-  ( ( ( # `  K
)  =  0  /\  N  =  0 )  ->  N  =  0 )
8382necon3ai 2688 . . . . . 6  |-  ( N  =/=  0  ->  -.  ( ( # `  K
)  =  0  /\  N  =  0 ) )
8481, 83syl 16 . . . . 5  |-  ( ph  ->  -.  ( ( # `  K )  =  0  /\  N  =  0 ) )
85 gcdn0cl 14000 . . . . 5  |-  ( ( ( ( # `  K
)  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( ( # `  K )  =  0  /\  N  =  0 ) )  ->  (
( # `  K )  gcd  N )  e.  NN )
8667, 69, 84, 85syl21anc 1222 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  e.  NN )
87 elnn1uz2 11147 . . . 4  |-  ( ( ( # `  K
)  gcd  N )  e.  NN  <->  ( ( (
# `  K )  gcd  N )  =  1  \/  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) ) )
8886, 87sylib 196 . . 3  |-  ( ph  ->  ( ( ( # `  K )  gcd  N
)  =  1  \/  ( ( # `  K
)  gcd  N )  e.  ( ZZ>= `  2 )
) )
8988ord 377 . 2  |-  ( ph  ->  ( -.  ( (
# `  K )  gcd  N )  =  1  ->  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) ) )
9080, 89mt3d 125 1  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   {crab 2811   _Vcvv 3106    C_ wss 3469   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Fincfn 7506   0cc0 9481   1c1 9482    x. cmul 9486   NNcn 10525   2c2 10574   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   #chash 12360    || cdivides 13836    gcd cgcd 13992   Primecprime 14065   Basecbs 14479   ↾s cress 14480   Grpcgrp 15716  SubGrpcsubg 15983   odcod 16338   Abelcabel 16588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-omul 7125  df-er 7301  df-ec 7303  df-qs 7307  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-acn 8312  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-dvds 13837  df-gcd 13993  df-prm 14066  df-pc 14209  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854  df-subg 15986  df-eqg 15988  df-ga 16116  df-od 16342  df-cmn 16589  df-abl 16590
This theorem is referenced by:  ablfacrp2  16901
  Copyright terms: Public domain W3C validator