MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablfacrp2 Unicode version

Theorem ablfacrp2 15137
Description: The factors  K ,  L of ablfacrp 15136 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 9897 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
4 ablfacrp.n . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
54nnnn0d 9897 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
63, 5nn0mulcld 9902 . . . . . . 7  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
71, 6eqeltrd 2327 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
8 ablfacrp.b . . . . . . . 8  |-  B  =  ( Base `  G
)
9 fvex 5391 . . . . . . . 8  |-  ( Base `  G )  e.  _V
108, 9eqeltri 2323 . . . . . . 7  |-  B  e. 
_V
11 hashclb 11230 . . . . . . 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 3179 . . . . . 6  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
1614, 15eqsstri 3129 . . . . 5  |-  K  C_  B
17 ssfi 6968 . . . . 5  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
1813, 16, 17sylancl 646 . . . 4  |-  ( ph  ->  K  e.  Fin )
19 hashcl 11228 . . . 4  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
2018, 19syl 17 . . 3  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
21 ablfacrp.g . . . . . . . 8  |-  ( ph  ->  G  e.  Abel )
222nnzd 9995 . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
23 ablfacrp.o . . . . . . . . 9  |-  O  =  ( od `  G
)
2423, 8oddvdssubg 14982 . . . . . . . 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 2337 . . . . . 6  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
278lagsubg 14514 . . . . . 6  |-  ( ( K  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 K )  ||  ( # `  B ) )
2826, 13, 27syl2anc 645 . . . . 5  |-  ( ph  ->  ( # `  K
)  ||  ( # `  B
) )
292nncnd 9642 . . . . . . 7  |-  ( ph  ->  M  e.  CC )
304nncnd 9642 . . . . . . 7  |-  ( ph  ->  N  e.  CC )
3129, 30mulcomd 8736 . . . . . 6  |-  ( ph  ->  ( M  x.  N
)  =  ( N  x.  M ) )
321, 31eqtrd 2285 . . . . 5  |-  ( ph  ->  ( # `  B
)  =  ( N  x.  M ) )
3328, 32breqtrd 3944 . . . 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 15135 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
3720nn0zd 9994 . . . . 5  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
384nnzd 9995 . . . . 5  |-  ( ph  ->  N  e.  ZZ )
39 coprmdvds 12655 . . . . 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 14982 . . . . . . . . . . 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 2337 . . . . . . . . 9  |-  ( ph  ->  L  e.  (SubGrp `  G ) )
458lagsubg 14514 . . . . . . . . 9  |-  ( ( L  e.  (SubGrp `  G )  /\  B  e.  Fin )  ->  ( # `
 L )  ||  ( # `  B ) )
4644, 13, 45syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  ||  ( # `  B
) )
4746, 1breqtrd 3944 . . . . . . 7  |-  ( ph  ->  ( # `  L
)  ||  ( M  x.  N ) )
48 gcdcom 12573 . . . . . . . . . 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 2287 . . . . . . . 8  |-  ( ph  ->  ( N  gcd  M
)  =  1 )
518, 23, 34, 14, 21, 4, 2, 50, 32ablfacrplem 15135 . . . . . . 7  |-  ( ph  ->  ( ( # `  L
)  gcd  M )  =  1 )
52 ssrab2 3179 . . . . . . . . . . . 12  |-  { x  e.  B  |  ( O `  x )  ||  N }  C_  B
5334, 52eqsstri 3129 . . . . . . . . . . 11  |-  L  C_  B
54 ssfi 6968 . . . . . . . . . . 11  |-  ( ( B  e.  Fin  /\  L  C_  B )  ->  L  e.  Fin )
5513, 53, 54sylancl 646 . . . . . . . . . 10  |-  ( ph  ->  L  e.  Fin )
56 hashcl 11228 . . . . . . . . . 10  |-  ( L  e.  Fin  ->  ( # `
 L )  e. 
NN0 )
5755, 56syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  L
)  e.  NN0 )
5857nn0zd 9994 . . . . . . . 8  |-  ( ph  ->  ( # `  L
)  e.  ZZ )
59 coprmdvds 12655 . . . . . . . 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 12429 . . . . . . 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 2253 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
66 eqid 2253 . . . . . . . . . 10  |-  ( LSSum `  G )  =  (
LSSum `  G )
678, 23, 14, 34, 21, 2, 4, 35, 1, 65, 66ablfacrp 15136 . . . . . . . . 9  |-  ( ph  ->  ( ( K  i^i  L )  =  { ( 0g `  G ) }  /\  ( K ( LSSum `  G ) L )  =  B ) )
6867simprd 451 . . . . . . . 8  |-  ( ph  ->  ( K ( LSSum `  G ) L )  =  B )
6968fveq2d 5381 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( # `  B
) )
70 eqid 2253 . . . . . . . 8  |-  (Cntz `  G )  =  (Cntz `  G )
7167simpld 447 . . . . . . . 8  |-  ( ph  ->  ( K  i^i  L
)  =  { ( 0g `  G ) } )
7270, 21, 26, 44ablcntzd 14984 . . . . . . . 8  |-  ( ph  ->  K  C_  ( (Cntz `  G ) `  L
) )
7366, 65, 70, 26, 44, 71, 72, 18, 55lsmhash 14849 . . . . . . 7  |-  ( ph  ->  ( # `  ( K ( LSSum `  G
) L ) )  =  ( ( # `  K )  x.  ( # `
 L ) ) )
7469, 73eqtr3d 2287 . . . . . 6  |-  ( ph  ->  ( # `  B
)  =  ( (
# `  K )  x.  ( # `  L
) ) )
7574, 1eqtr3d 2287 . . . . 5  |-  ( ph  ->  ( ( # `  K
)  x.  ( # `  L ) )  =  ( M  x.  N
) )
7664, 75breqtrrd 3946 . . . 4  |-  ( ph  ->  ( M  x.  ( # `
 L ) ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
7765subg0cl 14464 . . . . . . . 8  |-  ( L  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  L
)
78 ne0i 3368 . . . . . . . 8  |-  ( ( 0g `  G )  e.  L  ->  L  =/=  (/) )
7944, 77, 783syl 20 . . . . . . 7  |-  ( ph  ->  L  =/=  (/) )
80 hashnncl 11232 . . . . . . . 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 9670 . . . . 5  |-  ( ph  ->  ( # `  L
)  =/=  0 )
84 dvdsmulcr 12432 . . . . 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 12450 . . 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 12430 . . . . . . 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 3946 . . . 4  |-  ( ph  ->  ( ( # `  K
)  x.  N ) 
||  ( ( # `  K )  x.  ( # `
 L ) ) )
9388, 2eqeltrd 2327 . . . . . 6  |-  ( ph  ->  ( # `  K
)  e.  NN )
9493nnne0d 9670 . . . . 5  |-  ( ph  ->  ( # `  K
)  =/=  0 )
95 dvdscmulr 12431 . . . . 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 12450 . . 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 2412   {crab 2512   _Vcvv 2727    i^i cin 3077    C_ wss 3078   (/)c0 3362   {csn 3544   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Fincfn 6749   0cc0 8617   1c1 8618    x. cmul 8622   NNcn 9626   NN0cn0 9844   ZZcz 9903   #chash 11215    || cdivides 12405    gcd cgcd 12559   Basecbs 13022   0gc0g 13274  SubGrpcsubg 14450  Cntzccntz 14626   odcod 14675   LSSumclsm 14780   Abelcabel 14925
This theorem is referenced by:  ablfac1a  15139
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-disj 3892  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6546  df-ec 6548  df-qs 6552  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-acn 7459  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-q 10196  df-rp 10234  df-fz 10661  df-fzo 10749  df-fl 10803  df-mod 10852  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-divides 12406  df-gcd 12560  df-prime 12633  df-pc 12764  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-0g 13278  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-mulg 14327  df-subg 14453  df-eqg 14455  df-ga 14579  df-cntz 14628  df-od 14679  df-lsm 14782  df-pj1 14783  df-cmn 14926  df-abl 14927
  Copyright terms: Public domain W3C validator