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Theorem ablfac1lem 17701
Description: Lemma for ablfac1b 17703. 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 3432 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  Prime )
4 prmnn 14625 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
53, 4syl 17 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  NN )
6 ablfac1.g . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
7 ablgrp 17435 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8 ablfac1.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
98grpbn0 16695 . . . . . . . . 9  |-  ( G  e.  Grp  ->  B  =/=  (/) )
106, 7, 93syl 18 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
11 ablfac1.f . . . . . . . . 9  |-  ( ph  ->  B  e.  Fin )
12 hashnncl 12547 . . . . . . . . 9  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
1311, 12syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
1410, 13mpbird 236 . . . . . . 7  |-  ( ph  ->  ( # `  B
)  e.  NN )
1514adantr 467 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  e.  NN )
163, 15pccld 14800 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( P  pCnt  ( # `  B
) )  e.  NN0 )
175, 16nnexpcld 12437 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
181, 17syl5eqel 2533 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  NN )
19 ablfac1.n . . . 4  |-  N  =  ( ( # `  B
)  /  M )
20 pcdvds 14813 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  ||  ( # `
 B ) )
213, 15, 20syl2anc 667 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  ||  ( # `
 B ) )
221, 21syl5eqbr 4436 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  M  ||  ( # `  B
) )
23 nndivdvds 14311 . . . . . 6  |-  ( ( ( # `  B
)  e.  NN  /\  M  e.  NN )  ->  ( M  ||  ( # `
 B )  <->  ( ( # `
 B )  /  M )  e.  NN ) )
2415, 18, 23syl2anc 667 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( M  ||  ( # `  B
)  <->  ( ( # `  B )  /  M
)  e.  NN ) )
2522, 24mpbid 214 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( # `  B )  /  M )  e.  NN )
2619, 25syl5eqel 2533 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  NN )
2718, 26jca 535 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  e.  NN  /\  N  e.  NN ) )
281oveq1i 6300 . . 3  |-  ( M  gcd  N )  =  ( ( P ^
( P  pCnt  ( # `
 B ) ) )  gcd  N )
29 pcndvds2 14817 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  -.  P  ||  ( ( # `  B )  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
303, 15, 29syl2anc 667 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  -.  P  ||  ( ( # `  B )  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
311oveq2i 6301 . . . . . . . 8  |-  ( (
# `  B )  /  M )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3219, 31eqtri 2473 . . . . . . 7  |-  N  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3332breq2i 4410 . . . . . 6  |-  ( P 
||  N  <->  P  ||  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) ) )
3430, 33sylnibr 307 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  -.  P  ||  N )
3526nnzd 11039 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  ZZ )
36 coprm 14657 . . . . . 6  |-  ( ( P  e.  Prime  /\  N  e.  ZZ )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
373, 35, 36syl2anc 667 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( -.  P  ||  N  <->  ( P  gcd  N )  =  1 ) )
3834, 37mpbid 214 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P  gcd  N )  =  1 )
39 prmz 14626 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
403, 39syl 17 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  ZZ )
41 rpexp1i 14673 . . . . 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 1268 . . . 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 2497 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  gcd  N )  =  1 )
4519oveq2i 6301 . . 3  |-  ( M  x.  N )  =  ( M  x.  (
( # `  B )  /  M ) )
4615nncnd 10625 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  e.  CC )
4718nncnd 10625 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  CC )
4818nnne0d 10654 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  =/=  0 )
4946, 47, 48divcan2d 10385 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( M  x.  ( ( # `
 B )  /  M ) )  =  ( # `  B
) )
5045, 49syl5req 2498 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  =  ( M  x.  N
) )
5127, 44, 503jca 1188 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   {crab 2741    C_ wss 3404   (/)c0 3731   class class class wbr 4402    |-> cmpt 4461   ` cfv 5582  (class class class)co 6290   Fincfn 7569   1c1 9540    x. cmul 9544    / cdiv 10269   NNcn 10609   NN0cn0 10869   ZZcz 10937   ^cexp 12272   #chash 12515    || cdvds 14305    gcd cgcd 14468   Primecprime 14622    pCnt cpc 14786   Basecbs 15121   Grpcgrp 16669   odcod 17165   Abelcabl 17431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11785  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-grp 16673  df-abl 17433
This theorem is referenced by:  ablfac1a  17702  ablfac1b  17703
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