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Theorem ablfacrp2 15297
Description: The factors  K ,  L of ablfacrp 15296 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 10014 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
4 ablfacrp.n . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
54nnnn0d 10014 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
63, 5nn0mulcld 10019 . . . . . . 7  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
71, 6eqeltrd 2359 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
8 ablfacrp.b . . . . . . . 8  |-  B  =  ( Base `  G
)
9 fvex 5500 . . . . . . . 8  |-  ( Base `  G )  e.  _V
108, 9eqeltri 2355 . . . . . . 7  |-  B  e. 
_V
11 hashclb 11347 . . . . . . 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 3260 . . . . . 6  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
1614, 15eqsstri 3210 . . . . 5  |-  K  C_  B
17 ssfi 7079 . . . . 5  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
1813, 16, 17sylancl 645 . . . 4  |-  ( ph  ->  K  e.  Fin )
19 hashcl 11345 . . . 4  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
2018, 19syl 17 . . 3  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
21 ablfacrp.g . . . . . . . 8  |-  ( ph  ->  G  e.  Abel )
222nnzd 10112 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
23 ablfacrp.o . . . . . . . . 9  |-  O  =  ( od `  G
)
2423, 8oddvdssubg 15142 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G ) )
2521, 22, 24syl2anc 644 . . . . . . 7  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G
) )
2614, 25syl5eqel 2369 . . . . . 6  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
278lagsubg 14674 . . . . . 6  |-  ( ( K  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 K )  ||  ( # `  B ) )
2826, 13, 27syl2anc 644 . . . . 5  |-  ( ph  ->  ( # `  K
)  ||  ( # `  B
) )
292nncnd 9758 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
304nncnd 9758 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
3129, 30mulcomd 8852 . . . . . 6  |-  ( ph  ->  ( M  x.  N
)  =  ( N  x.  M ) )
321, 31eqtrd 2317 . . . . 5  |-  ( ph  ->  ( # `  B
)  =  ( N  x.  M ) )
3328, 32breqtrd 4049 . . . 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 15295 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
3720nn0zd 10111 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
384nnzd 10112 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
39 coprmdvds 12776 . . . . 5  |-  ( ( ( # `  K
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( ( # `  K
)  ||  ( N  x.  M )  /\  (
( # `  K )  gcd  N )  =  1 )  ->  ( # `
 K )  ||  M ) )
4037, 38, 22, 39syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( ( # `  K )  ||  ( N  x.  M )  /\  ( ( # `  K
)  gcd  N )  =  1 )  -> 
( # `  K ) 
||  M ) )
4133, 36, 40mp2and 662 . . 3  |-  ( ph  ->  ( # `  K
)  ||  M )
4223, 8oddvdssubg 15142 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  N  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G ) )
4321, 38, 42syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G
) )
4434, 43syl5eqel 2369 . . . . . . . . 9  |-  ( ph  ->  L  e.  (SubGrp `  G ) )
458lagsubg 14674 . . . . . . . . 9  |-  ( ( L  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 L )  ||  ( # `  B ) )
4644, 13, 45syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  ||  ( # `  B
) )
4746, 1breqtrd 4049 . . . . . . 7  |-  ( ph  ->  ( # `  L
)  ||  ( M  x.  N ) )
48 gcdcom 12694 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  gcd  N
)  =  ( N  gcd  M ) )
4922, 38, 48syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( M  gcd  N
)  =  ( N  gcd  M ) )
5049, 35eqtr3d 2319 . . . . . . . 8  |-  ( ph  ->  ( N  gcd  M
)  =  1 )
518, 23, 34, 14, 21, 4, 2, 50, 32ablfacrplem 15295 . . . . . . 7  |-  ( ph  ->  ( ( # `  L
)  gcd  M )  =  1 )
52 ssrab2 3260 . . . . . . . . . . . 12  |-  { x  e.  B  |  ( O `  x )  ||  N }  C_  B
5334, 52eqsstri 3210 . . . . . . . . . . 11  |-  L  C_  B
54 ssfi 7079 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  L  C_  B )  ->  L  e.  Fin )
5513, 53, 54sylancl 645 . . . . . . . . . 10  |-  ( ph  ->  L  e.  Fin )
56 hashcl 11345 . . . . . . . . . 10  |-  ( L  e.  Fin  ->  ( # `
 L )  e. 
NN0 )
5755, 56syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  L
)  e.  NN0 )
5857nn0zd 10111 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  e.  ZZ )
59 coprmdvds 12776 . . . . . . . 8  |-  ( ( ( # `  L
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( ( # `  L
)  ||  ( M  x.  N )  /\  (
( # `  L )  gcd  M )  =  1 )  ->  ( # `
 L )  ||  N ) )
6058, 22, 38, 59syl3anc 1184 . . . . . . 7  |-  ( ph  ->  ( ( ( # `  L )  ||  ( M  x.  N )  /\  ( ( # `  L
)  gcd  M )  =  1 )  -> 
( # `  L ) 
||  N ) )
6147, 51, 60mp2and 662 . . . . . 6  |-  ( ph  ->  ( # `  L
)  ||  N )
62 dvdscmul 12550 . . . . . . 7  |-  ( ( ( # `  L
)  e.  ZZ  /\  N  e.  ZZ  /\  M  e.  ZZ )  ->  (
( # `  L ) 
||  N  ->  ( M  x.  ( # `  L
) )  ||  ( M  x.  N )
) )
6358, 38, 22, 62syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( # `  L
)  ||  N  ->  ( M  x.  ( # `  L ) )  ||  ( M  x.  N
) ) )
6461, 63mpd 16 . . . . 5  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( M  x.  N ) )
65 eqid 2285 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
66 eqid 2285 . . . . . . . . . 10  |-  ( LSSum `  G )  =  (
LSSum `  G )
678, 23, 14, 34, 21, 2, 4, 35, 1, 65, 66ablfacrp 15296 . . . . . . . . 9  |-  ( ph  ->  ( ( K  i^i  L )  =  { ( 0g `  G ) }  /\  ( K ( LSSum `  G ) L )  =  B ) )
6867simprd 451 . . . . . . . 8  |-  ( ph  ->  ( K ( LSSum `  G ) L )  =  B )
6968fveq2d 5490 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( # `  B
) )
70 eqid 2285 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
7167simpld 447 . . . . . . . 8  |-  ( ph  ->  ( K  i^i  L
)  =  { ( 0g `  G ) } )
7270, 21, 26, 44ablcntzd 15144 . . . . . . . 8  |-  ( ph  ->  K  C_  ( (Cntz `  G ) `  L
) )
7366, 65, 70, 26, 44, 71, 72, 18, 55lsmhash 15009 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( ( # `  K )  x.  ( # `
 L ) ) )
7469, 73eqtr3d 2319 . . . . . 6  |-  ( ph  ->  ( # `  B
)  =  ( (
# `  K )  x.  ( # `  L
) ) )
7574, 1eqtr3d 2319 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  ( # `  L ) )  =  ( M  x.  N
) )
7664, 75breqtrrd 4051 . . . 4  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
7765subg0cl 14624 . . . . . . . 8  |-  ( L  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  L
)
78 ne0i 3463 . . . . . . . 8  |-  ( ( 0g `  G )  e.  L  ->  L  =/=  (/) )
7944, 77, 783syl 20 . . . . . . 7  |-  ( ph  ->  L  =/=  (/) )
80 hashnncl 11349 . . . . . . . 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 9786 . . . . 5  |-  ( ph  ->  ( # `  L
)  =/=  0 )
84 dvdsmulcr 12553 . . . . 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 1188 . . . 4  |-  ( ph  ->  ( ( M  x.  ( # `  L ) )  ||  ( (
# `  K )  x.  ( # `  L
) )  <->  M  ||  ( # `
 K ) ) )
8676, 85mpbid 203 . . 3  |-  ( ph  ->  M  ||  ( # `  K ) )
87 dvdseq 12571 . . 3  |-  ( ( ( ( # `  K
)  e.  NN0  /\  M  e.  NN0 )  /\  ( ( # `  K
)  ||  M  /\  M  ||  ( # `  K
) ) )  -> 
( # `  K )  =  M )
8820, 3, 41, 86, 87syl22anc 1185 . 2  |-  ( ph  ->  ( # `  K
)  =  M )
89 dvdsmulc 12551 . . . . . . 7  |-  ( ( ( # `  K
)  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( # `  K ) 
||  M  ->  (
( # `  K )  x.  N )  ||  ( M  x.  N
) ) )
9037, 22, 38, 89syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( # `  K
)  ||  M  ->  ( ( # `  K
)  x.  N ) 
||  ( M  x.  N ) ) )
9141, 90mpd 16 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( M  x.  N ) )
9291, 75breqtrrd 4051 . . . 4  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
9388, 2eqeltrd 2359 . . . . . 6  |-  ( ph  ->  ( # `  K
)  e.  NN )
9493nnne0d 9786 . . . . 5  |-  ( ph  ->  ( # `  K
)  =/=  0 )
95 dvdscmulr 12552 . . . . 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 1188 . . . 4  |-  ( ph  ->  ( ( ( # `  K )  x.  N
)  ||  ( ( # `
 K )  x.  ( # `  L
) )  <->  N  ||  ( # `
 L ) ) )
9792, 96mpbid 203 . . 3  |-  ( ph  ->  N  ||  ( # `  L ) )
98 dvdseq 12571 . . 3  |-  ( ( ( ( # `  L
)  e.  NN0  /\  N  e.  NN0 )  /\  ( ( # `  L
)  ||  N  /\  N  ||  ( # `  L
) ) )  -> 
( # `  L )  =  N )
9957, 5, 61, 97, 98syl22anc 1185 . 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 1624    e. wcel 1685    =/= wne 2448   {crab 2549   _Vcvv 2790    i^i cin 3153    C_ wss 3154   (/)c0 3457   {csn 3642   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Fincfn 6859   0cc0 8733   1c1 8734    x. cmul 8738   NNcn 9742   NN0cn0 9961   ZZcz 10020   #chash 11332    || cdivides 12526    gcd cgcd 12680   Basecbs 13143   0gc0g 13395  SubGrpcsubg 14610  Cntzccntz 14786   odcod 14835   LSSumclsm 14940   Abelcabel 15085
This theorem is referenced by:  ablfac1a  15299
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-disj 3996  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-omul 6480  df-er 6656  df-ec 6658  df-qs 6662  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-acn 7571  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-q 10313  df-rp 10351  df-fz 10778  df-fzo 10866  df-fl 10920  df-mod 10969  df-seq 11042  df-exp 11100  df-fac 11284  df-bc 11311  df-hash 11333  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-clim 11957  df-sum 12154  df-dvds 12527  df-gcd 12681  df-prm 12754  df-pc 12885  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-0g 13399  df-mnd 14362  df-submnd 14411  df-grp 14484  df-minusg 14485  df-sbg 14486  df-mulg 14487  df-subg 14613  df-eqg 14615  df-ga 14739  df-cntz 14788  df-od 14839  df-lsm 14942  df-pj1 14943  df-cmn 15086  df-abl 15087
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