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Theorem ablfac1lem 15319
Description: Lemma for ablfac1b 15321. 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 3193 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  Prime )
4 prmnn 12777 . . . . . 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 15110 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8 ablfac1.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
98grpbn0 14527 . . . . . . . . 9  |-  ( G  e.  Grp  ->  B  =/=  (/) )
106, 7, 93syl 18 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
11 ablfac1.f . . . . . . . . 9  |-  ( ph  ->  B  e.  Fin )
12 hashnncl 11370 . . . . . . . . 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 12919 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( P  pCnt  ( # `  B
) )  e.  NN0 )
175, 16nnexpcld 11282 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
181, 17syl5eqel 2380 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  NN )
19 ablfac1.n . . . 4  |-  N  =  ( ( # `  B
)  /  M )
20 pcdvds 12932 . . . . . . 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 4072 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  M  ||  ( # `  B
) )
23 nndivdvds 12553 . . . . . 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 2380 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  NN )
2718, 26jca 518 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  e.  NN  /\  N  e.  NN ) )
281oveq1i 5884 . . 3  |-  ( M  gcd  N )  =  ( ( P ^
( P  pCnt  ( # `
 B ) ) )  gcd  N )
29 pcndvds2 12936 . . . . . . 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 5885 . . . . . . . 8  |-  ( (
# `  B )  /  M )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3219, 31eqtri 2316 . . . . . . 7  |-  N  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3332breq2i 4047 . . . . . 6  |-  ( P 
||  N  <->  P  ||  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) ) )
3430, 33sylnibr 296 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  -.  P  ||  N )
3526nnzd 10132 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  ZZ )
36 coprm 12795 . . . . . 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 12778 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
403, 39syl 15 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  ZZ )
41 rpexp1i 12816 . . . . 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 2340 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  gcd  N )  =  1 )
4519oveq2i 5885 . . 3  |-  ( M  x.  N )  =  ( M  x.  (
( # `  B )  /  M ) )
4615nncnd 9778 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  e.  CC )
4718nncnd 9778 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  CC )
4818nnne0d 9806 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  =/=  0 )
4946, 47, 48divcan2d 9554 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( M  x.  ( ( # `
 B )  /  M ) )  =  ( # `  B
) )
5045, 49syl5req 2341 . 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 1632    e. wcel 1696    =/= wne 2459   {crab 2560    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Fincfn 6879   1c1 8754    x. cmul 8758    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120   #chash 11353    || cdivides 12547    gcd cgcd 12701   Primecprime 12774    pCnt cpc 12905   Basecbs 13164   Grpcgrp 14378   odcod 14856   Abelcabel 15106
This theorem is referenced by:  ablfac1a  15320  ablfac1b  15321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-0g 13420  df-mnd 14383  df-grp 14505  df-abl 15108
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