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Theorem ablfacrplem 15263
Description: Lemma for ablfacrp2 15265. (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
StepHypRef Expression
1 nprmdvds1 12753 . . . . . . 7  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
21adantl 454 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  1 )
3 ablfacrp.1 . . . . . . . 8  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
43adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( M  gcd  N )  =  1 )
54breq2d 4009 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( M  gcd  N
)  <->  p  ||  1 ) )
62, 5mtbird 294 . . . . 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 10084 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ZZ )
11 ablfacrp.o . . . . . . . . . . . . . . . 16  |-  O  =  ( od `  G
)
12 ablfacrp.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  G
)
1311, 12oddvdssubg 15110 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G ) )
148, 10, 13syl2anc 645 . . . . . . . . . . . . . 14  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G
) )
157, 14syl5eqel 2342 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
1615ad2antrr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  e.  (SubGrp `  G )
)
17 eqid 2258 . . . . . . . . . . . . 13  |-  ( Gs  K )  =  ( Gs  K )
1817subggrp 14587 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  ( Gs  K
)  e.  Grp )
1916, 18syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( Gs  K )  e.  Grp )
2017subgbas 14588 . . . . . . . . . . . . 13  |-  ( K  e.  (SubGrp `  G
)  ->  K  =  ( Base `  ( Gs  K
) ) )
2116, 20syl 17 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  =  ( Base `  ( Gs  K ) ) )
22 ablfacrp.2 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( # `  B
)  =  ( M  x.  N ) )
239nnnn0d 9986 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  M  e.  NN0 )
24 ablfacrp.n . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  NN )
2524nnnn0d 9986 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  NN0 )
2623, 25nn0mulcld 9991 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
2722, 26eqeltrd 2332 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
28 fvex 5472 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
2912, 28eqeltri 2328 . . . . . . . . . . . . . . . 16  |-  B  e. 
_V
30 hashclb 11319 . . . . . . . . . . . . . . . 16  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
3129, 30ax-mp 10 . . . . . . . . . . . . . . 15  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
3227, 31sylibr 205 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  Fin )
33 ssrab2 3233 . . . . . . . . . . . . . . 15  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
347, 33eqsstri 3183 . . . . . . . . . . . . . 14  |-  K  C_  B
35 ssfi 7051 . . . . . . . . . . . . . 14  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
3632, 34, 35sylancl 646 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Fin )
3736ad2antrr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  e.  Fin )
3821, 37eqeltrrd 2333 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( Base `  ( Gs  K ) )  e.  Fin )
39 simplr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  e.  Prime )
40 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  ( # `  K
) )
4121fveq2d 5462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( # `
 K )  =  ( # `  ( Base `  ( Gs  K ) ) ) )
4240, 41breqtrd 4021 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  ( # `  ( Base `  ( Gs  K ) ) ) )
43 eqid 2258 . . . . . . . . . . . 12  |-  ( Base `  ( Gs  K ) )  =  ( Base `  ( Gs  K ) )
44 eqid 2258 . . . . . . . . . . . 12  |-  ( od
`  ( Gs  K ) )  =  ( od
`  ( Gs  K ) )
4543, 44odcau 14878 . . . . . . . . . . 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 1190 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  E. g  e.  ( Base `  ( Gs  K ) ) ( ( od `  ( Gs  K ) ) `  g )  =  p )
4721rexeqdv 2718 . . . . . . . . . 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 225 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p )
4917, 11, 44subgod 14844 . . . . . . . . . . . . 13  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  K )  ->  ( O `  g )  =  ( ( od
`  ( Gs  K ) ) `  g ) )
5016, 49sylan 459 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  ( O `  g )  =  ( ( od
`  ( Gs  K ) ) `  g ) )
51 fveq2 5458 . . . . . . . . . . . . . . . 16  |-  ( x  =  g  ->  ( O `  x )  =  ( O `  g ) )
5251breq1d 4007 . . . . . . . . . . . . . . 15  |-  ( x  =  g  ->  (
( O `  x
)  ||  M  <->  ( O `  g )  ||  M
) )
5352, 7elrab2 2900 . . . . . . . . . . . . . 14  |-  ( g  e.  K  <->  ( g  e.  B  /\  ( O `  g )  ||  M ) )
5453simprbi 452 . . . . . . . . . . . . 13  |-  ( g  e.  K  ->  ( O `  g )  ||  M )
5554adantl 454 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  ( O `  g )  ||  M )
5650, 55eqbrtrrd 4019 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  (
( od `  ( Gs  K ) ) `  g )  ||  M
)
57 breq1 4000 . . . . . . . . . . 11  |-  ( ( ( od `  ( Gs  K ) ) `  g )  =  p  ->  ( ( ( od `  ( Gs  K ) ) `  g
)  ||  M  <->  p  ||  M
) )
5856, 57syl5ibcom 213 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  (
( ( od `  ( Gs  K ) ) `  g )  =  p  ->  p  ||  M
) )
5958rexlimdva 2642 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p  ->  p  ||  M
) )
6048, 59mpd 16 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  M )
6160ex 425 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( # `  K
)  ->  p  ||  M
) )
6261anim1d 549 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  ( # `  K
)  /\  p  ||  N
)  ->  ( p  ||  M  /\  p  ||  N ) ) )
63 prmz 12724 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  ZZ )
6463adantl 454 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  p  e.  ZZ )
65 hashcl 11317 . . . . . . . . . 10  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
6636, 65syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
6766nn0zd 10083 . . . . . . . 8  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
6867adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( # `  K
)  e.  ZZ )
6924nnzd 10084 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
7069adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  N  e.  ZZ )
71 dvdsgcdb 12686 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( # `  K )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( p  ||  ( # `
 K )  /\  p  ||  N )  <->  p  ||  (
( # `  K )  gcd  N ) ) )
7264, 68, 70, 71syl3anc 1187 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  ( # `  K
)  /\  p  ||  N
)  <->  p  ||  ( (
# `  K )  gcd  N ) ) )
7310adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  M  e.  ZZ )
74 dvdsgcdb 12686 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( p  ||  M  /\  p  ||  N )  <-> 
p  ||  ( M  gcd  N ) ) )
7564, 73, 70, 74syl3anc 1187 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  M  /\  p  ||  N )  <->  p  ||  ( M  gcd  N ) ) )
7662, 72, 753imtr3d 260 . . . . 5  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( ( # `  K
)  gcd  N )  ->  p  ||  ( M  gcd  N ) ) )
776, 76mtod 170 . . . 4  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  ( ( # `  K
)  gcd  N )
)
7877nrexdv 2621 . . 3  |-  ( ph  ->  -.  E. p  e. 
Prime  p  ||  ( (
# `  K )  gcd  N ) )
79 exprmfct 12752 . . 3  |-  ( ( ( # `  K
)  gcd  N )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( ( # `  K )  gcd  N
) )
8078, 79nsyl 115 . 2  |-  ( ph  ->  -.  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) )
8124nnne0d 9758 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
82 simpr 449 . . . . . . 7  |-  ( ( ( # `  K
)  =  0  /\  N  =  0 )  ->  N  =  0 )
8382necon3ai 2461 . . . . . 6  |-  ( N  =/=  0  ->  -.  ( ( # `  K
)  =  0  /\  N  =  0 ) )
8481, 83syl 17 . . . . 5  |-  ( ph  ->  -.  ( ( # `  K )  =  0  /\  N  =  0 ) )
85 gcdn0cl 12656 . . . . 5  |-  ( ( ( ( # `  K
)  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( ( # `  K )  =  0  /\  N  =  0 ) )  ->  (
( # `  K )  gcd  N )  e.  NN )
8667, 69, 84, 85syl21anc 1186 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  e.  NN )
87 elnn1uz2 10262 . . . 4  |-  ( ( ( # `  K
)  gcd  N )  e.  NN  <->  ( ( (
# `  K )  gcd  N )  =  1  \/  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) ) )
8886, 87sylib 190 . . 3  |-  ( ph  ->  ( ( ( # `  K )  gcd  N
)  =  1  \/  ( ( # `  K
)  gcd  N )  e.  ( ZZ>= `  2 )
) )
8988ord 368 . 2  |-  ( ph  ->  ( -.  ( (
# `  K )  gcd  N )  =  1  ->  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) ) )
9080, 89mt3d 119 1  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   {crab 2522   _Vcvv 2763    C_ wss 3127   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Fincfn 6831   0cc0 8705   1c1 8706    x. cmul 8710   NNcn 9714   2c2 9763   NN0cn0 9933   ZZcz 9992   ZZ>=cuz 10198   #chash 11304    || cdivides 12494    gcd cgcd 12648   Primecprime 12721   Basecbs 13111   ↾s cress 13112   Grpcgrp 14325  SubGrpcsubg 14578   odcod 14803   Abelcabel 15053
This theorem is referenced by:  ablfacrp2  15265
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-disj 3968  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-omul 6452  df-er 6628  df-ec 6630  df-qs 6634  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-acn 7543  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-q 10285  df-rp 10323  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-fac 11256  df-bc 11283  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-sum 12125  df-divides 12495  df-gcd 12649  df-prime 12722  df-pc 12853  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-0g 13367  df-mnd 14330  df-submnd 14379  df-grp 14452  df-minusg 14453  df-sbg 14454  df-mulg 14455  df-subg 14581  df-eqg 14583  df-ga 14707  df-od 14807  df-cmn 15054  df-abl 15055
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