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Theorem ablfac1lem 16921
Description: Lemma for ablfac1b 16923. 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 3504 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  Prime )
4 prmnn 14079 . . . . . 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 16609 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8 ablfac1.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
98grpbn0 15889 . . . . . . . . 9  |-  ( G  e.  Grp  ->  B  =/=  (/) )
106, 7, 93syl 20 . . . . . . . 8  |-  ( ph  ->  B  =/=  (/) )
11 ablfac1.f . . . . . . . . 9  |-  ( ph  ->  B  e.  Fin )
12 hashnncl 12404 . . . . . . . . 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 14233 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  ( P  pCnt  ( # `  B
) )  e.  NN0 )
175, 16nnexpcld 12299 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
181, 17syl5eqel 2559 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  NN )
19 ablfac1.n . . . 4  |-  N  =  ( ( # `  B
)  /  M )
20 pcdvds 14246 . . . . . . 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 4480 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  M  ||  ( # `  B
) )
23 nndivdvds 13853 . . . . . 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 2559 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  NN )
2718, 26jca 532 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  e.  NN  /\  N  e.  NN ) )
281oveq1i 6294 . . 3  |-  ( M  gcd  N )  =  ( ( P ^
( P  pCnt  ( # `
 B ) ) )  gcd  N )
29 pcndvds2 14250 . . . . . . 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 6295 . . . . . . . 8  |-  ( (
# `  B )  /  M )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3219, 31eqtri 2496 . . . . . . 7  |-  N  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3332breq2i 4455 . . . . . 6  |-  ( P 
||  N  <->  P  ||  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) ) )
3430, 33sylnibr 305 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  -.  P  ||  N )
3526nnzd 10965 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  N  e.  ZZ )
36 coprm 14100 . . . . . 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 14080 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ZZ )
403, 39syl 16 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  P  e.  ZZ )
41 rpexp1i 14121 . . . . 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 1228 . . . 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 2520 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( M  gcd  N )  =  1 )
4519oveq2i 6295 . . 3  |-  ( M  x.  N )  =  ( M  x.  (
( # `  B )  /  M ) )
4615nncnd 10552 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  e.  CC )
4718nncnd 10552 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  e.  CC )
4818nnne0d 10580 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  M  =/=  0 )
4946, 47, 48divcan2d 10322 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( M  x.  ( ( # `
 B )  /  M ) )  =  ( # `  B
) )
5045, 49syl5req 2521 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  =  ( M  x.  N
) )
5127, 44, 503jca 1176 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818    C_ wss 3476   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   Fincfn 7516   1c1 9493    x. cmul 9497    / cdiv 10206   NNcn 10536   NN0cn0 10795   ZZcz 10864   ^cexp 12134   #chash 12373    || cdivides 13847    gcd cgcd 14003   Primecprime 14076    pCnt cpc 14219   Basecbs 14490   Grpcgrp 15727   odcod 16355   Abelcabl 16605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-fz 11673  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-prm 14077  df-pc 14220  df-0g 14697  df-mnd 15732  df-grp 15867  df-abl 16607
This theorem is referenced by:  ablfac1a  16922  ablfac1b  16923
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