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Theorem ablfacrp2 15250
Description: The factors  K ,  L of ablfacrp 15249 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 9971 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
4 ablfacrp.n . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
54nnnn0d 9971 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
63, 5nn0mulcld 9976 . . . . . . 7  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
71, 6eqeltrd 2330 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
8 ablfacrp.b . . . . . . . 8  |-  B  =  ( Base `  G
)
9 fvex 5458 . . . . . . . 8  |-  ( Base `  G )  e.  _V
108, 9eqeltri 2326 . . . . . . 7  |-  B  e. 
_V
11 hashclb 11304 . . . . . . 7  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
1210, 11ax-mp 10 . . . . . 6  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
137, 12sylibr 205 . . . . 5  |-  ( ph  ->  B  e.  Fin )
14 ablfacrp.k . . . . . 6  |-  K  =  { x  e.  B  |  ( O `  x )  ||  M }
15 ssrab2 3219 . . . . . 6  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
1614, 15eqsstri 3169 . . . . 5  |-  K  C_  B
17 ssfi 7037 . . . . 5  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
1813, 16, 17sylancl 646 . . . 4  |-  ( ph  ->  K  e.  Fin )
19 hashcl 11302 . . . 4  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
2018, 19syl 17 . . 3  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
21 ablfacrp.g . . . . . . . 8  |-  ( ph  ->  G  e.  Abel )
222nnzd 10069 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
23 ablfacrp.o . . . . . . . . 9  |-  O  =  ( od `  G
)
2423, 8oddvdssubg 15095 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G ) )
2521, 22, 24syl2anc 645 . . . . . . 7  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G
) )
2614, 25syl5eqel 2340 . . . . . 6  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
278lagsubg 14627 . . . . . 6  |-  ( ( K  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 K )  ||  ( # `  B ) )
2826, 13, 27syl2anc 645 . . . . 5  |-  ( ph  ->  ( # `  K
)  ||  ( # `  B
) )
292nncnd 9716 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
304nncnd 9716 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
3129, 30mulcomd 8810 . . . . . 6  |-  ( ph  ->  ( M  x.  N
)  =  ( N  x.  M ) )
321, 31eqtrd 2288 . . . . 5  |-  ( ph  ->  ( # `  B
)  =  ( N  x.  M ) )
3328, 32breqtrd 4007 . . . 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 15248 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
3720nn0zd 10068 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
384nnzd 10069 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
39 coprmdvds 12729 . . . . 5  |-  ( ( ( # `  K
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( ( # `  K
)  ||  ( N  x.  M )  /\  (
( # `  K )  gcd  N )  =  1 )  ->  ( # `
 K )  ||  M ) )
4037, 38, 22, 39syl3anc 1187 . . . 4  |-  ( ph  ->  ( ( ( # `  K )  ||  ( N  x.  M )  /\  ( ( # `  K
)  gcd  N )  =  1 )  -> 
( # `  K ) 
||  M ) )
4133, 36, 40mp2and 663 . . 3  |-  ( ph  ->  ( # `  K
)  ||  M )
4223, 8oddvdssubg 15095 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  N  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G ) )
4321, 38, 42syl2anc 645 . . . . . . . . . 10  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G
) )
4434, 43syl5eqel 2340 . . . . . . . . 9  |-  ( ph  ->  L  e.  (SubGrp `  G ) )
458lagsubg 14627 . . . . . . . . 9  |-  ( ( L  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 L )  ||  ( # `  B ) )
4644, 13, 45syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  ||  ( # `  B
) )
4746, 1breqtrd 4007 . . . . . . 7  |-  ( ph  ->  ( # `  L
)  ||  ( M  x.  N ) )
48 gcdcom 12647 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  ( N  gcd  M ) )
4922, 38, 48syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  ( M  gcd  N
)  =  ( N  gcd  M ) )
5049, 35eqtr3d 2290 . . . . . . . 8  |-  ( ph  ->  ( N  gcd  M
)  =  1 )
518, 23, 34, 14, 21, 4, 2, 50, 32ablfacrplem 15248 . . . . . . 7  |-  ( ph  ->  ( ( # `  L
)  gcd  M )  =  1 )
52 ssrab2 3219 . . . . . . . . . . . 12  |-  { x  e.  B  |  ( O `  x )  ||  N }  C_  B
5334, 52eqsstri 3169 . . . . . . . . . . 11  |-  L  C_  B
54 ssfi 7037 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  L  C_  B )  ->  L  e.  Fin )
5513, 53, 54sylancl 646 . . . . . . . . . 10  |-  ( ph  ->  L  e.  Fin )
56 hashcl 11302 . . . . . . . . . 10  |-  ( L  e.  Fin  ->  ( # `
 L )  e. 
NN0 )
5755, 56syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  L
)  e.  NN0 )
5857nn0zd 10068 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  e.  ZZ )
59 coprmdvds 12729 . . . . . . . 8  |-  ( ( ( # `  L
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( # `  L
)  ||  ( M  x.  N )  /\  (
( # `  L )  gcd  M )  =  1 )  ->  ( # `
 L )  ||  N ) )
6058, 22, 38, 59syl3anc 1187 . . . . . . 7  |-  ( ph  ->  ( ( ( # `  L )  ||  ( M  x.  N )  /\  ( ( # `  L
)  gcd  M )  =  1 )  -> 
( # `  L ) 
||  N ) )
6147, 51, 60mp2and 663 . . . . . 6  |-  ( ph  ->  ( # `  L
)  ||  N )
62 dvdscmul 12503 . . . . . . 7  |-  ( ( ( # `  L
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( # `  L ) 
||  N  ->  ( M  x.  ( # `  L
) )  ||  ( M  x.  N )
) )
6358, 38, 22, 62syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( ( # `  L
)  ||  N  ->  ( M  x.  ( # `  L ) )  ||  ( M  x.  N
) ) )
6461, 63mpd 16 . . . . 5  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( M  x.  N ) )
65 eqid 2256 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
66 eqid 2256 . . . . . . . . . 10  |-  ( LSSum `  G )  =  (
LSSum `  G )
678, 23, 14, 34, 21, 2, 4, 35, 1, 65, 66ablfacrp 15249 . . . . . . . . 9  |-  ( ph  ->  ( ( K  i^i  L )  =  { ( 0g `  G ) }  /\  ( K ( LSSum `  G ) L )  =  B ) )
6867simprd 451 . . . . . . . 8  |-  ( ph  ->  ( K ( LSSum `  G ) L )  =  B )
6968fveq2d 5448 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( # `  B
) )
70 eqid 2256 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
7167simpld 447 . . . . . . . 8  |-  ( ph  ->  ( K  i^i  L
)  =  { ( 0g `  G ) } )
7270, 21, 26, 44ablcntzd 15097 . . . . . . . 8  |-  ( ph  ->  K  C_  ( (Cntz `  G ) `  L
) )
7366, 65, 70, 26, 44, 71, 72, 18, 55lsmhash 14962 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( ( # `  K )  x.  ( # `
 L ) ) )
7469, 73eqtr3d 2290 . . . . . 6  |-  ( ph  ->  ( # `  B
)  =  ( (
# `  K )  x.  ( # `  L
) ) )
7574, 1eqtr3d 2290 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  ( # `  L ) )  =  ( M  x.  N
) )
7664, 75breqtrrd 4009 . . . 4  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
7765subg0cl 14577 . . . . . . . 8  |-  ( L  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  L
)
78 ne0i 3422 . . . . . . . 8  |-  ( ( 0g `  G )  e.  L  ->  L  =/=  (/) )
7944, 77, 783syl 20 . . . . . . 7  |-  ( ph  ->  L  =/=  (/) )
80 hashnncl 11306 . . . . . . . 8  |-  ( L  e.  Fin  ->  (
( # `  L )  e.  NN  <->  L  =/=  (/) ) )
8155, 80syl 17 . . . . . . 7  |-  ( ph  ->  ( ( # `  L
)  e.  NN  <->  L  =/=  (/) ) )
8279, 81mpbird 225 . . . . . 6  |-  ( ph  ->  ( # `  L
)  e.  NN )
8382nnne0d 9744 . . . . 5  |-  ( ph  ->  ( # `  L
)  =/=  0 )
84 dvdsmulcr 12506 . . . . 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 1191 . . . 4  |-  ( ph  ->  ( ( M  x.  ( # `  L ) )  ||  ( (
# `  K )  x.  ( # `  L
) )  <->  M  ||  ( # `
 K ) ) )
8676, 85mpbid 203 . . 3  |-  ( ph  ->  M  ||  ( # `  K ) )
87 dvdseq 12524 . . 3  |-  ( ( ( ( # `  K
)  e.  NN0  /\  M  e.  NN0 )  /\  ( ( # `  K
)  ||  M  /\  M  ||  ( # `  K
) ) )  -> 
( # `  K )  =  M )
8820, 3, 41, 86, 87syl22anc 1188 . 2  |-  ( ph  ->  ( # `  K
)  =  M )
89 dvdsmulc 12504 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( # `  K ) 
||  M  ->  (
( # `  K )  x.  N )  ||  ( M  x.  N
) ) )
9037, 22, 38, 89syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  M  ->  ( ( # `  K
)  x.  N ) 
||  ( M  x.  N ) ) )
9141, 90mpd 16 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( M  x.  N ) )
9291, 75breqtrrd 4009 . . . 4  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
9388, 2eqeltrd 2330 . . . . . 6  |-  ( ph  ->  ( # `  K
)  e.  NN )
9493nnne0d 9744 . . . . 5  |-  ( ph  ->  ( # `  K
)  =/=  0 )
95 dvdscmulr 12505 . . . . 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 1191 . . . 4  |-  ( ph  ->  ( ( ( # `  K )  x.  N
)  ||  ( ( # `
 K )  x.  ( # `  L
) )  <->  N  ||  ( # `
 L ) ) )
9792, 96mpbid 203 . . 3  |-  ( ph  ->  N  ||  ( # `  L ) )
98 dvdseq 12524 . . 3  |-  ( ( ( ( # `  L
)  e.  NN0  /\  N  e.  NN0 )  /\  ( ( # `  L
)  ||  N  /\  N  ||  ( # `  L
) ) )  -> 
( # `  L )  =  N )
9957, 5, 61, 97, 98syl22anc 1188 . 2  |-  ( ph  ->  ( # `  L
)  =  N )
10088, 99jca 520 1  |-  ( ph  ->  ( ( # `  K
)  =  M  /\  ( # `  L )  =  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   {crab 2520   _Vcvv 2757    i^i cin 3112    C_ wss 3113   (/)c0 3416   {csn 3600   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Fincfn 6817   0cc0 8691   1c1 8692    x. cmul 8696   NNcn 9700   NN0cn0 9918   ZZcz 9977   #chash 11289    || cdivides 12479    gcd cgcd 12633   Basecbs 13096   0gc0g 13348  SubGrpcsubg 14563  Cntzccntz 14739   odcod 14788   LSSumclsm 14893   Abelcabel 15038
This theorem is referenced by:  ablfac1a  15252
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-disj 3954  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-omul 6438  df-er 6614  df-ec 6616  df-qs 6620  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-oi 7179  df-card 7526  df-acn 7529  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-q 10270  df-rp 10308  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-clim 11913  df-sum 12110  df-divides 12480  df-gcd 12634  df-prime 12707  df-pc 12838  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-0g 13352  df-mnd 14315  df-submnd 14364  df-grp 14437  df-minusg 14438  df-sbg 14439  df-mulg 14440  df-subg 14566  df-eqg 14568  df-ga 14692  df-cntz 14741  df-od 14792  df-lsm 14895  df-pj1 14896  df-cmn 15039  df-abl 15040
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