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Theorem ablfac1lem 15303
Description: Lemma for ablfac1b 15305. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
ablfac1.b  |-  B  =  ( Base `  G
)
ablfac1.o  |-  O  =  ( od `  G
)
ablfac1.s  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
ablfac1.g  |-  ( ph  ->  G  e.  Abel )
ablfac1.f  |-  ( ph  ->  B  e.  Fin )
ablfac1.1  |-  ( ph  ->  A  C_  Prime )
ablfac1.m  |-  M  =  ( P ^ ( P  pCnt  ( # `  B
) ) )
ablfac1.n  |-  N  =  ( ( # `  B
)  /  M )
Assertion
Ref Expression
ablfac1lem  |-  ( (
ph  /\  P  e.  A )  ->  (
( M  e.  NN  /\  N  e.  NN )  /\  ( M  gcd  N )  =  1  /\  ( # `  B
)  =  ( M  x.  N ) ) )
Distinct variable groups:    x, p, B    ph, p, x    A, p, x    O, p, x    P, p, x    G, p, x
Allowed substitution hints:    S( x, p)    M( x, p)    N( x, p)

Proof of Theorem ablfac1lem
StepHypRef Expression
1 ablfac1.m . . . 4  |-  M  =  ( P ^ ( P  pCnt  ( # `  B
) ) )
2 ablfac1.1 . . . . . . 7  |-  ( ph  ->  A  C_  Prime )
32sselda 3180 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  Prime )
4 prmnn 12761 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
53, 4syl 15 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  NN )
6 ablfac1.g . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
7 ablgrp 15094 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8 ablfac1.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
98grpbn0 14511 . . . . . . . . 9  |-  ( G  e.  Grp  ->  B  =/=  (/) )
106, 7, 93syl 18 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
11 ablfac1.f . . . . . . . . 9  |-  ( ph  ->  B  e.  Fin )
12 hashnncl 11354 . . . . . . . . 9  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
1311, 12syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
1410, 13mpbird 223 . . . . . . 7  |-  ( ph  ->  ( # `  B
)  e.  NN )
1514adantr 451 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  e.  NN )
163, 15pccld 12903 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( P  pCnt  ( # `  B
) )  e.  NN0 )
175, 16nnexpcld 11266 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
181, 17syl5eqel 2367 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  NN )
19 ablfac1.n . . . 4  |-  N  =  ( ( # `  B
)  /  M )
20 pcdvds 12916 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  ||  ( # `
 B ) )
213, 15, 20syl2anc 642 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  ||  ( # `
 B ) )
221, 21syl5eqbr 4056 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  M  ||  ( # `  B
) )
23 nndivdvds 12537 . . . . . 6  |-  ( ( ( # `  B
)  e.  NN  /\  M  e.  NN )  ->  ( M  ||  ( # `
 B )  <->  ( ( # `
 B )  /  M )  e.  NN ) )
2415, 18, 23syl2anc 642 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( M  ||  ( # `  B
)  <->  ( ( # `  B )  /  M
)  e.  NN ) )
2522, 24mpbid 201 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( # `  B )  /  M )  e.  NN )
2619, 25syl5eqel 2367 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  NN )
2718, 26jca 518 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  e.  NN  /\  N  e.  NN ) )
281oveq1i 5868 . . 3  |-  ( M  gcd  N )  =  ( ( P ^
( P  pCnt  ( # `
 B ) ) )  gcd  N )
29 pcndvds2 12920 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  -.  P  ||  ( ( # `  B )  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
303, 15, 29syl2anc 642 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  -.  P  ||  ( ( # `  B )  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
311oveq2i 5869 . . . . . . . 8  |-  ( (
# `  B )  /  M )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3219, 31eqtri 2303 . . . . . . 7  |-  N  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3332breq2i 4031 . . . . . 6  |-  ( P 
||  N  <->  P  ||  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) ) )
3430, 33sylnibr 296 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  -.  P  ||  N )
3526nnzd 10116 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  ZZ )
36 coprm 12779 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
373, 35, 36syl2anc 642 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
3834, 37mpbid 201 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P  gcd  N )  =  1 )
39 prmz 12762 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
403, 39syl 15 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  ZZ )
41 rpexp1i 12800 . . . . 5  |-  ( ( P  e.  ZZ  /\  N  e.  ZZ  /\  ( P  pCnt  ( # `  B
) )  e.  NN0 )  ->  ( ( P  gcd  N )  =  1  ->  ( ( P ^ ( P  pCnt  (
# `  B )
) )  gcd  N
)  =  1 ) )
4240, 35, 16, 41syl3anc 1182 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( P  gcd  N
)  =  1  -> 
( ( P ^
( P  pCnt  ( # `
 B ) ) )  gcd  N )  =  1 ) )
4338, 42mpd 14 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  gcd 
N )  =  1 )
4428, 43syl5eq 2327 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  gcd  N )  =  1 )
4519oveq2i 5869 . . 3  |-  ( M  x.  N )  =  ( M  x.  (
( # `  B )  /  M ) )
4615nncnd 9762 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  e.  CC )
4718nncnd 9762 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  CC )
4818nnne0d 9790 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  =/=  0 )
4946, 47, 48divcan2d 9538 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( M  x.  ( ( # `
 B )  /  M ) )  =  ( # `  B
) )
5045, 49syl5req 2328 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  =  ( M  x.  N
) )
5127, 44, 503jca 1132 1  |-  ( (
ph  /\  P  e.  A )  ->  (
( M  e.  NN  /\  N  e.  NN )  /\  ( M  gcd  N )  =  1  /\  ( # `  B
)  =  ( M  x.  N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    C_ wss 3152   (/)c0 3455   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   Fincfn 6863   1c1 8738    x. cmul 8742    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ^cexp 11104   #chash 11337    || cdivides 12531    gcd cgcd 12685   Primecprime 12758    pCnt cpc 12889   Basecbs 13148   Grpcgrp 14362   odcod 14840   Abelcabel 15090
This theorem is referenced by:  ablfac1a  15304  ablfac1b  15305
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  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-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  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-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-0g 13404  df-mnd 14367  df-grp 14489  df-abl 15092
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