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Theorem ablfacrp2 17330
Description: The factors  K ,  L of ablfacrp 17329 have the expected orders (which allows for repeated application to decompose  G into subgroups of prime-power order). Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablfacrp.b  |-  B  =  ( Base `  G
)
ablfacrp.o  |-  O  =  ( od `  G
)
ablfacrp.k  |-  K  =  { x  e.  B  |  ( O `  x )  ||  M }
ablfacrp.l  |-  L  =  { x  e.  B  |  ( O `  x )  ||  N }
ablfacrp.g  |-  ( ph  ->  G  e.  Abel )
ablfacrp.m  |-  ( ph  ->  M  e.  NN )
ablfacrp.n  |-  ( ph  ->  N  e.  NN )
ablfacrp.1  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
ablfacrp.2  |-  ( ph  ->  ( # `  B
)  =  ( M  x.  N ) )
Assertion
Ref Expression
ablfacrp2  |-  ( ph  ->  ( ( # `  K
)  =  M  /\  ( # `  L )  =  N ) )
Distinct variable groups:    x, B    x, G    x, O    x, M    x, N    ph, x
Allowed substitution hints:    K( x)    L( x)

Proof of Theorem ablfacrp2
StepHypRef Expression
1 ablfacrp.2 . . . . . . 7  |-  ( ph  ->  ( # `  B
)  =  ( M  x.  N ) )
2 ablfacrp.m . . . . . . . . 9  |-  ( ph  ->  M  e.  NN )
32nnnn0d 10813 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
4 ablfacrp.n . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
54nnnn0d 10813 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
63, 5nn0mulcld 10818 . . . . . . 7  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
71, 6eqeltrd 2490 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
8 ablfacrp.b . . . . . . . 8  |-  B  =  ( Base `  G
)
9 fvex 5815 . . . . . . . 8  |-  ( Base `  G )  e.  _V
108, 9eqeltri 2486 . . . . . . 7  |-  B  e. 
_V
11 hashclb 12384 . . . . . . 7  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
1210, 11ax-mp 5 . . . . . 6  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
137, 12sylibr 212 . . . . 5  |-  ( ph  ->  B  e.  Fin )
14 ablfacrp.k . . . . . 6  |-  K  =  { x  e.  B  |  ( O `  x )  ||  M }
15 ssrab2 3523 . . . . . 6  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
1614, 15eqsstri 3471 . . . . 5  |-  K  C_  B
17 ssfi 7695 . . . . 5  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
1813, 16, 17sylancl 660 . . . 4  |-  ( ph  ->  K  e.  Fin )
19 hashcl 12382 . . . 4  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
2018, 19syl 17 . . 3  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
21 ablfacrp.g . . . . . . . 8  |-  ( ph  ->  G  e.  Abel )
222nnzd 10927 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
23 ablfacrp.o . . . . . . . . 9  |-  O  =  ( od `  G
)
2423, 8oddvdssubg 17077 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G ) )
2521, 22, 24syl2anc 659 . . . . . . 7  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G
) )
2614, 25syl5eqel 2494 . . . . . 6  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
278lagsubg 16479 . . . . . 6  |-  ( ( K  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 K )  ||  ( # `  B ) )
2826, 13, 27syl2anc 659 . . . . 5  |-  ( ph  ->  ( # `  K
)  ||  ( # `  B
) )
292nncnd 10512 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
304nncnd 10512 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
3129, 30mulcomd 9567 . . . . . 6  |-  ( ph  ->  ( M  x.  N
)  =  ( N  x.  M ) )
321, 31eqtrd 2443 . . . . 5  |-  ( ph  ->  ( # `  B
)  =  ( N  x.  M ) )
3328, 32breqtrd 4418 . . . 4  |-  ( ph  ->  ( # `  K
)  ||  ( N  x.  M ) )
34 ablfacrp.l . . . . 5  |-  L  =  { x  e.  B  |  ( O `  x )  ||  N }
35 ablfacrp.1 . . . . 5  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
368, 23, 14, 34, 21, 2, 4, 35, 1ablfacrplem 17328 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
3720nn0zd 10926 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
384nnzd 10927 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
39 coprmdvds 14344 . . . . 5  |-  ( ( ( # `  K
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( ( # `  K
)  ||  ( N  x.  M )  /\  (
( # `  K )  gcd  N )  =  1 )  ->  ( # `
 K )  ||  M ) )
4037, 38, 22, 39syl3anc 1230 . . . 4  |-  ( ph  ->  ( ( ( # `  K )  ||  ( N  x.  M )  /\  ( ( # `  K
)  gcd  N )  =  1 )  -> 
( # `  K ) 
||  M ) )
4133, 36, 40mp2and 677 . . 3  |-  ( ph  ->  ( # `  K
)  ||  M )
4223, 8oddvdssubg 17077 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  N  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G ) )
4321, 38, 42syl2anc 659 . . . . . . . . . 10  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G
) )
4434, 43syl5eqel 2494 . . . . . . . . 9  |-  ( ph  ->  L  e.  (SubGrp `  G ) )
458lagsubg 16479 . . . . . . . . 9  |-  ( ( L  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 L )  ||  ( # `  B ) )
4644, 13, 45syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  ||  ( # `  B
) )
4746, 1breqtrd 4418 . . . . . . 7  |-  ( ph  ->  ( # `  L
)  ||  ( M  x.  N ) )
48 gcdcom 14259 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  ( N  gcd  M ) )
4922, 38, 48syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( M  gcd  N
)  =  ( N  gcd  M ) )
5049, 35eqtr3d 2445 . . . . . . . 8  |-  ( ph  ->  ( N  gcd  M
)  =  1 )
518, 23, 34, 14, 21, 4, 2, 50, 32ablfacrplem 17328 . . . . . . 7  |-  ( ph  ->  ( ( # `  L
)  gcd  M )  =  1 )
52 ssrab2 3523 . . . . . . . . . . . 12  |-  { x  e.  B  |  ( O `  x )  ||  N }  C_  B
5334, 52eqsstri 3471 . . . . . . . . . . 11  |-  L  C_  B
54 ssfi 7695 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  L  C_  B )  ->  L  e.  Fin )
5513, 53, 54sylancl 660 . . . . . . . . . 10  |-  ( ph  ->  L  e.  Fin )
56 hashcl 12382 . . . . . . . . . 10  |-  ( L  e.  Fin  ->  ( # `
 L )  e. 
NN0 )
5755, 56syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  L
)  e.  NN0 )
5857nn0zd 10926 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  e.  ZZ )
59 coprmdvds 14344 . . . . . . . 8  |-  ( ( ( # `  L
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( # `  L
)  ||  ( M  x.  N )  /\  (
( # `  L )  gcd  M )  =  1 )  ->  ( # `
 L )  ||  N ) )
6058, 22, 38, 59syl3anc 1230 . . . . . . 7  |-  ( ph  ->  ( ( ( # `  L )  ||  ( M  x.  N )  /\  ( ( # `  L
)  gcd  M )  =  1 )  -> 
( # `  L ) 
||  N ) )
6147, 51, 60mp2and 677 . . . . . 6  |-  ( ph  ->  ( # `  L
)  ||  N )
62 dvdscmul 14111 . . . . . . 7  |-  ( ( ( # `  L
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( # `  L ) 
||  N  ->  ( M  x.  ( # `  L
) )  ||  ( M  x.  N )
) )
6358, 38, 22, 62syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( ( # `  L
)  ||  N  ->  ( M  x.  ( # `  L ) )  ||  ( M  x.  N
) ) )
6461, 63mpd 15 . . . . 5  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( M  x.  N ) )
65 eqid 2402 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
66 eqid 2402 . . . . . . . . . 10  |-  ( LSSum `  G )  =  (
LSSum `  G )
678, 23, 14, 34, 21, 2, 4, 35, 1, 65, 66ablfacrp 17329 . . . . . . . . 9  |-  ( ph  ->  ( ( K  i^i  L )  =  { ( 0g `  G ) }  /\  ( K ( LSSum `  G ) L )  =  B ) )
6867simprd 461 . . . . . . . 8  |-  ( ph  ->  ( K ( LSSum `  G ) L )  =  B )
6968fveq2d 5809 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( # `  B
) )
70 eqid 2402 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
7167simpld 457 . . . . . . . 8  |-  ( ph  ->  ( K  i^i  L
)  =  { ( 0g `  G ) } )
7270, 21, 26, 44ablcntzd 17079 . . . . . . . 8  |-  ( ph  ->  K  C_  ( (Cntz `  G ) `  L
) )
7366, 65, 70, 26, 44, 71, 72, 18, 55lsmhash 16939 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( ( # `  K )  x.  ( # `
 L ) ) )
7469, 73eqtr3d 2445 . . . . . 6  |-  ( ph  ->  ( # `  B
)  =  ( (
# `  K )  x.  ( # `  L
) ) )
7574, 1eqtr3d 2445 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  ( # `  L ) )  =  ( M  x.  N
) )
7664, 75breqtrrd 4420 . . . 4  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
7765subg0cl 16425 . . . . . . . 8  |-  ( L  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  L
)
78 ne0i 3743 . . . . . . . 8  |-  ( ( 0g `  G )  e.  L  ->  L  =/=  (/) )
7944, 77, 783syl 20 . . . . . . 7  |-  ( ph  ->  L  =/=  (/) )
80 hashnncl 12391 . . . . . . . 8  |-  ( L  e.  Fin  ->  (
( # `  L )  e.  NN  <->  L  =/=  (/) ) )
8155, 80syl 17 . . . . . . 7  |-  ( ph  ->  ( ( # `  L
)  e.  NN  <->  L  =/=  (/) ) )
8279, 81mpbird 232 . . . . . 6  |-  ( ph  ->  ( # `  L
)  e.  NN )
8382nnne0d 10541 . . . . 5  |-  ( ph  ->  ( # `  L
)  =/=  0 )
84 dvdsmulcr 14114 . . . . 5  |-  ( ( M  e.  ZZ  /\  ( # `  K )  e.  ZZ  /\  (
( # `  L )  e.  ZZ  /\  ( # `
 L )  =/=  0 ) )  -> 
( ( M  x.  ( # `  L ) )  ||  ( (
# `  K )  x.  ( # `  L
) )  <->  M  ||  ( # `
 K ) ) )
8522, 37, 58, 83, 84syl112anc 1234 . . . 4  |-  ( ph  ->  ( ( M  x.  ( # `  L ) )  ||  ( (
# `  K )  x.  ( # `  L
) )  <->  M  ||  ( # `
 K ) ) )
8676, 85mpbid 210 . . 3  |-  ( ph  ->  M  ||  ( # `  K ) )
87 dvdseq 14134 . . 3  |-  ( ( ( ( # `  K
)  e.  NN0  /\  M  e.  NN0 )  /\  ( ( # `  K
)  ||  M  /\  M  ||  ( # `  K
) ) )  -> 
( # `  K )  =  M )
8820, 3, 41, 86, 87syl22anc 1231 . 2  |-  ( ph  ->  ( # `  K
)  =  M )
89 dvdsmulc 14112 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( # `  K ) 
||  M  ->  (
( # `  K )  x.  N )  ||  ( M  x.  N
) ) )
9037, 22, 38, 89syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  M  ->  ( ( # `  K
)  x.  N ) 
||  ( M  x.  N ) ) )
9141, 90mpd 15 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( M  x.  N ) )
9291, 75breqtrrd 4420 . . . 4  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
9388, 2eqeltrd 2490 . . . . . 6  |-  ( ph  ->  ( # `  K
)  e.  NN )
9493nnne0d 10541 . . . . 5  |-  ( ph  ->  ( # `  K
)  =/=  0 )
95 dvdscmulr 14113 . . . . 5  |-  ( ( N  e.  ZZ  /\  ( # `  L )  e.  ZZ  /\  (
( # `  K )  e.  ZZ  /\  ( # `
 K )  =/=  0 ) )  -> 
( ( ( # `  K )  x.  N
)  ||  ( ( # `
 K )  x.  ( # `  L
) )  <->  N  ||  ( # `
 L ) ) )
9638, 58, 37, 94, 95syl112anc 1234 . . . 4  |-  ( ph  ->  ( ( ( # `  K )  x.  N
)  ||  ( ( # `
 K )  x.  ( # `  L
) )  <->  N  ||  ( # `
 L ) ) )
9792, 96mpbid 210 . . 3  |-  ( ph  ->  N  ||  ( # `  L ) )
98 dvdseq 14134 . . 3  |-  ( ( ( ( # `  L
)  e.  NN0  /\  N  e.  NN0 )  /\  ( ( # `  L
)  ||  N  /\  N  ||  ( # `  L
) ) )  -> 
( # `  L )  =  N )
9957, 5, 61, 97, 98syl22anc 1231 . 2  |-  ( ph  ->  ( # `  L
)  =  N )
10088, 99jca 530 1  |-  ( ph  ->  ( ( # `  K
)  =  M  /\  ( # `  L )  =  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2757   _Vcvv 3058    i^i cin 3412    C_ wss 3413   (/)c0 3737   {csn 3971   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Fincfn 7474   0cc0 9442   1c1 9443    x. cmul 9447   NNcn 10496   NN0cn0 10756   ZZcz 10825   #chash 12359    || cdvds 14087    gcd cgcd 14245   Basecbs 14733   0gc0g 14946  SubGrpcsubg 16411  Cntzccntz 16569   odcod 16765   LSSumclsm 16870   Abelcabl 17015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-omul 7092  df-er 7268  df-ec 7270  df-qs 7274  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-acn 8275  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-q 11146  df-rp 11184  df-fz 11644  df-fzo 11768  df-fl 11879  df-mod 11948  df-seq 12062  df-exp 12121  df-fac 12308  df-bc 12335  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-sum 13565  df-dvds 14088  df-gcd 14246  df-prm 14319  df-pc 14462  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-0g 14948  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-submnd 16183  df-grp 16273  df-minusg 16274  df-sbg 16275  df-mulg 16276  df-subg 16414  df-eqg 16416  df-ga 16544  df-cntz 16571  df-od 16769  df-lsm 16872  df-pj1 16873  df-cmn 17016  df-abl 17017
This theorem is referenced by:  ablfac1a  17332
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