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Theorem ablfacrp2 15318
Description: The factors  K ,  L of ablfacrp 15317 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 10034 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
4 ablfacrp.n . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
54nnnn0d 10034 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
63, 5nn0mulcld 10039 . . . . . . 7  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
71, 6eqeltrd 2370 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
8 ablfacrp.b . . . . . . . 8  |-  B  =  ( Base `  G
)
9 fvex 5555 . . . . . . . 8  |-  ( Base `  G )  e.  _V
108, 9eqeltri 2366 . . . . . . 7  |-  B  e. 
_V
11 hashclb 11368 . . . . . . 7  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
1210, 11ax-mp 8 . . . . . 6  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
137, 12sylibr 203 . . . . 5  |-  ( ph  ->  B  e.  Fin )
14 ablfacrp.k . . . . . 6  |-  K  =  { x  e.  B  |  ( O `  x )  ||  M }
15 ssrab2 3271 . . . . . 6  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
1614, 15eqsstri 3221 . . . . 5  |-  K  C_  B
17 ssfi 7099 . . . . 5  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
1813, 16, 17sylancl 643 . . . 4  |-  ( ph  ->  K  e.  Fin )
19 hashcl 11366 . . . 4  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
2018, 19syl 15 . . 3  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
21 ablfacrp.g . . . . . . . 8  |-  ( ph  ->  G  e.  Abel )
222nnzd 10132 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
23 ablfacrp.o . . . . . . . . 9  |-  O  =  ( od `  G
)
2423, 8oddvdssubg 15163 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G ) )
2521, 22, 24syl2anc 642 . . . . . . 7  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G
) )
2614, 25syl5eqel 2380 . . . . . 6  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
278lagsubg 14695 . . . . . 6  |-  ( ( K  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 K )  ||  ( # `  B ) )
2826, 13, 27syl2anc 642 . . . . 5  |-  ( ph  ->  ( # `  K
)  ||  ( # `  B
) )
292nncnd 9778 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
304nncnd 9778 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
3129, 30mulcomd 8872 . . . . . 6  |-  ( ph  ->  ( M  x.  N
)  =  ( N  x.  M ) )
321, 31eqtrd 2328 . . . . 5  |-  ( ph  ->  ( # `  B
)  =  ( N  x.  M ) )
3328, 32breqtrd 4063 . . . 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 15316 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
3720nn0zd 10131 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
384nnzd 10132 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
39 coprmdvds 12797 . . . . 5  |-  ( ( ( # `  K
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( ( # `  K
)  ||  ( N  x.  M )  /\  (
( # `  K )  gcd  N )  =  1 )  ->  ( # `
 K )  ||  M ) )
4037, 38, 22, 39syl3anc 1182 . . . 4  |-  ( ph  ->  ( ( ( # `  K )  ||  ( N  x.  M )  /\  ( ( # `  K
)  gcd  N )  =  1 )  -> 
( # `  K ) 
||  M ) )
4133, 36, 40mp2and 660 . . 3  |-  ( ph  ->  ( # `  K
)  ||  M )
4223, 8oddvdssubg 15163 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  N  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G ) )
4321, 38, 42syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G
) )
4434, 43syl5eqel 2380 . . . . . . . . 9  |-  ( ph  ->  L  e.  (SubGrp `  G ) )
458lagsubg 14695 . . . . . . . . 9  |-  ( ( L  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 L )  ||  ( # `  B ) )
4644, 13, 45syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  ||  ( # `  B
) )
4746, 1breqtrd 4063 . . . . . . 7  |-  ( ph  ->  ( # `  L
)  ||  ( M  x.  N ) )
48 gcdcom 12715 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  ( N  gcd  M ) )
4922, 38, 48syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( M  gcd  N
)  =  ( N  gcd  M ) )
5049, 35eqtr3d 2330 . . . . . . . 8  |-  ( ph  ->  ( N  gcd  M
)  =  1 )
518, 23, 34, 14, 21, 4, 2, 50, 32ablfacrplem 15316 . . . . . . 7  |-  ( ph  ->  ( ( # `  L
)  gcd  M )  =  1 )
52 ssrab2 3271 . . . . . . . . . . . 12  |-  { x  e.  B  |  ( O `  x )  ||  N }  C_  B
5334, 52eqsstri 3221 . . . . . . . . . . 11  |-  L  C_  B
54 ssfi 7099 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  L  C_  B )  ->  L  e.  Fin )
5513, 53, 54sylancl 643 . . . . . . . . . 10  |-  ( ph  ->  L  e.  Fin )
56 hashcl 11366 . . . . . . . . . 10  |-  ( L  e.  Fin  ->  ( # `
 L )  e. 
NN0 )
5755, 56syl 15 . . . . . . . . 9  |-  ( ph  ->  ( # `  L
)  e.  NN0 )
5857nn0zd 10131 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  e.  ZZ )
59 coprmdvds 12797 . . . . . . . 8  |-  ( ( ( # `  L
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( # `  L
)  ||  ( M  x.  N )  /\  (
( # `  L )  gcd  M )  =  1 )  ->  ( # `
 L )  ||  N ) )
6058, 22, 38, 59syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( ( # `  L )  ||  ( M  x.  N )  /\  ( ( # `  L
)  gcd  M )  =  1 )  -> 
( # `  L ) 
||  N ) )
6147, 51, 60mp2and 660 . . . . . 6  |-  ( ph  ->  ( # `  L
)  ||  N )
62 dvdscmul 12571 . . . . . . 7  |-  ( ( ( # `  L
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( # `  L ) 
||  N  ->  ( M  x.  ( # `  L
) )  ||  ( M  x.  N )
) )
6358, 38, 22, 62syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( # `  L
)  ||  N  ->  ( M  x.  ( # `  L ) )  ||  ( M  x.  N
) ) )
6461, 63mpd 14 . . . . 5  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( M  x.  N ) )
65 eqid 2296 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
66 eqid 2296 . . . . . . . . . 10  |-  ( LSSum `  G )  =  (
LSSum `  G )
678, 23, 14, 34, 21, 2, 4, 35, 1, 65, 66ablfacrp 15317 . . . . . . . . 9  |-  ( ph  ->  ( ( K  i^i  L )  =  { ( 0g `  G ) }  /\  ( K ( LSSum `  G ) L )  =  B ) )
6867simprd 449 . . . . . . . 8  |-  ( ph  ->  ( K ( LSSum `  G ) L )  =  B )
6968fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( # `  B
) )
70 eqid 2296 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
7167simpld 445 . . . . . . . 8  |-  ( ph  ->  ( K  i^i  L
)  =  { ( 0g `  G ) } )
7270, 21, 26, 44ablcntzd 15165 . . . . . . . 8  |-  ( ph  ->  K  C_  ( (Cntz `  G ) `  L
) )
7366, 65, 70, 26, 44, 71, 72, 18, 55lsmhash 15030 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( ( # `  K )  x.  ( # `
 L ) ) )
7469, 73eqtr3d 2330 . . . . . 6  |-  ( ph  ->  ( # `  B
)  =  ( (
# `  K )  x.  ( # `  L
) ) )
7574, 1eqtr3d 2330 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  ( # `  L ) )  =  ( M  x.  N
) )
7664, 75breqtrrd 4065 . . . 4  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
7765subg0cl 14645 . . . . . . . 8  |-  ( L  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  L
)
78 ne0i 3474 . . . . . . . 8  |-  ( ( 0g `  G )  e.  L  ->  L  =/=  (/) )
7944, 77, 783syl 18 . . . . . . 7  |-  ( ph  ->  L  =/=  (/) )
80 hashnncl 11370 . . . . . . . 8  |-  ( L  e.  Fin  ->  (
( # `  L )  e.  NN  <->  L  =/=  (/) ) )
8155, 80syl 15 . . . . . . 7  |-  ( ph  ->  ( ( # `  L
)  e.  NN  <->  L  =/=  (/) ) )
8279, 81mpbird 223 . . . . . 6  |-  ( ph  ->  ( # `  L
)  e.  NN )
8382nnne0d 9806 . . . . 5  |-  ( ph  ->  ( # `  L
)  =/=  0 )
84 dvdsmulcr 12574 . . . . 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 1186 . . . 4  |-  ( ph  ->  ( ( M  x.  ( # `  L ) )  ||  ( (
# `  K )  x.  ( # `  L
) )  <->  M  ||  ( # `
 K ) ) )
8676, 85mpbid 201 . . 3  |-  ( ph  ->  M  ||  ( # `  K ) )
87 dvdseq 12592 . . 3  |-  ( ( ( ( # `  K
)  e.  NN0  /\  M  e.  NN0 )  /\  ( ( # `  K
)  ||  M  /\  M  ||  ( # `  K
) ) )  -> 
( # `  K )  =  M )
8820, 3, 41, 86, 87syl22anc 1183 . 2  |-  ( ph  ->  ( # `  K
)  =  M )
89 dvdsmulc 12572 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( # `  K ) 
||  M  ->  (
( # `  K )  x.  N )  ||  ( M  x.  N
) ) )
9037, 22, 38, 89syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  M  ->  ( ( # `  K
)  x.  N ) 
||  ( M  x.  N ) ) )
9141, 90mpd 14 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( M  x.  N ) )
9291, 75breqtrrd 4065 . . . 4  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
9388, 2eqeltrd 2370 . . . . . 6  |-  ( ph  ->  ( # `  K
)  e.  NN )
9493nnne0d 9806 . . . . 5  |-  ( ph  ->  ( # `  K
)  =/=  0 )
95 dvdscmulr 12573 . . . . 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 1186 . . . 4  |-  ( ph  ->  ( ( ( # `  K )  x.  N
)  ||  ( ( # `
 K )  x.  ( # `  L
) )  <->  N  ||  ( # `
 L ) ) )
9792, 96mpbid 201 . . 3  |-  ( ph  ->  N  ||  ( # `  L ) )
98 dvdseq 12592 . . 3  |-  ( ( ( ( # `  L
)  e.  NN0  /\  N  e.  NN0 )  /\  ( ( # `  L
)  ||  N  /\  N  ||  ( # `  L
) ) )  -> 
( # `  L )  =  N )
9957, 5, 61, 97, 98syl22anc 1183 . 2  |-  ( ph  ->  ( # `  L
)  =  N )
10088, 99jca 518 1  |-  ( ph  ->  ( ( # `  K
)  =  M  /\  ( # `  L )  =  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Fincfn 6879   0cc0 8753   1c1 8754    x. cmul 8758   NNcn 9762   NN0cn0 9981   ZZcz 10040   #chash 11353    || cdivides 12547    gcd cgcd 12701   Basecbs 13164   0gc0g 13416  SubGrpcsubg 14631  Cntzccntz 14807   odcod 14856   LSSumclsm 14961   Abelcabel 15106
This theorem is referenced by:  ablfac1a  15320
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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-disj 4010  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-se 4369  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-isom 5280  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-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  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-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-eqg 14636  df-ga 14760  df-cntz 14809  df-od 14860  df-lsm 14963  df-pj1 14964  df-cmn 15107  df-abl 15108
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