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Theorem ablfac1lem 16574
Description: Lemma for ablfac1b 16576. 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 3361 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  Prime )
4 prmnn 13771 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
53, 4syl 16 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  NN )
6 ablfac1.g . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
7 ablgrp 16287 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8 ablfac1.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
98grpbn0 15572 . . . . . . . . 9  |-  ( G  e.  Grp  ->  B  =/=  (/) )
106, 7, 93syl 20 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
11 ablfac1.f . . . . . . . . 9  |-  ( ph  ->  B  e.  Fin )
12 hashnncl 12139 . . . . . . . . 9  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
1311, 12syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
1410, 13mpbird 232 . . . . . . 7  |-  ( ph  ->  ( # `  B
)  e.  NN )
1514adantr 465 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  e.  NN )
163, 15pccld 13922 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( P  pCnt  ( # `  B
) )  e.  NN0 )
175, 16nnexpcld 12034 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
181, 17syl5eqel 2527 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  NN )
19 ablfac1.n . . . 4  |-  N  =  ( ( # `  B
)  /  M )
20 pcdvds 13935 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  ||  ( # `
 B ) )
213, 15, 20syl2anc 661 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  ||  ( # `
 B ) )
221, 21syl5eqbr 4330 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  M  ||  ( # `  B
) )
23 nndivdvds 13546 . . . . . 6  |-  ( ( ( # `  B
)  e.  NN  /\  M  e.  NN )  ->  ( M  ||  ( # `
 B )  <->  ( ( # `
 B )  /  M )  e.  NN ) )
2415, 18, 23syl2anc 661 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( M  ||  ( # `  B
)  <->  ( ( # `  B )  /  M
)  e.  NN ) )
2522, 24mpbid 210 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( # `  B )  /  M )  e.  NN )
2619, 25syl5eqel 2527 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  NN )
2718, 26jca 532 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  e.  NN  /\  N  e.  NN ) )
281oveq1i 6106 . . 3  |-  ( M  gcd  N )  =  ( ( P ^
( P  pCnt  ( # `
 B ) ) )  gcd  N )
29 pcndvds2 13939 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  -.  P  ||  ( ( # `  B )  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
303, 15, 29syl2anc 661 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  -.  P  ||  ( ( # `  B )  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
311oveq2i 6107 . . . . . . . 8  |-  ( (
# `  B )  /  M )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3219, 31eqtri 2463 . . . . . . 7  |-  N  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3332breq2i 4305 . . . . . 6  |-  ( P 
||  N  <->  P  ||  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) ) )
3430, 33sylnibr 305 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  -.  P  ||  N )
3526nnzd 10751 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  ZZ )
36 coprm 13791 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
373, 35, 36syl2anc 661 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
3834, 37mpbid 210 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P  gcd  N )  =  1 )
39 prmz 13772 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
403, 39syl 16 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  ZZ )
41 rpexp1i 13812 . . . . 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 1218 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( P  gcd  N
)  =  1  -> 
( ( P ^
( P  pCnt  ( # `
 B ) ) )  gcd  N )  =  1 ) )
4338, 42mpd 15 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  gcd 
N )  =  1 )
4428, 43syl5eq 2487 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  gcd  N )  =  1 )
4519oveq2i 6107 . . 3  |-  ( M  x.  N )  =  ( M  x.  (
( # `  B )  /  M ) )
4615nncnd 10343 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  e.  CC )
4718nncnd 10343 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  CC )
4818nnne0d 10371 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  =/=  0 )
4946, 47, 48divcan2d 10114 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( M  x.  ( ( # `
 B )  /  M ) )  =  ( # `  B
) )
5045, 49syl5req 2488 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  =  ( M  x.  N
) )
5127, 44, 503jca 1168 1  |-  ( (
ph  /\  P  e.  A )  ->  (
( M  e.  NN  /\  N  e.  NN )  /\  ( M  gcd  N )  =  1  /\  ( # `  B
)  =  ( M  x.  N ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   {crab 2724    C_ wss 3333   (/)c0 3642   class class class wbr 4297    e. cmpt 4355   ` cfv 5423  (class class class)co 6096   Fincfn 7315   1c1 9288    x. cmul 9292    / cdiv 9998   NNcn 10327   NN0cn0 10584   ZZcz 10651   ^cexp 11870   #chash 12108    || cdivides 13540    gcd cgcd 13695   Primecprime 13768    pCnt cpc 13908   Basecbs 14179   Grpcgrp 15415   odcod 16033   Abelcabel 16283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-0g 14385  df-mnd 15420  df-grp 15550  df-abl 16285
This theorem is referenced by:  ablfac1a  16575  ablfac1b  16576
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