MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablfac1c Structured version   Unicode version

Theorem ablfac1c 16584
Description: The factors of ablfac1b 16583 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
ablfac1.b  |-  B  =  ( Base `  G
)
ablfac1.o  |-  O  =  ( od `  G
)
ablfac1.s  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
ablfac1.g  |-  ( ph  ->  G  e.  Abel )
ablfac1.f  |-  ( ph  ->  B  e.  Fin )
ablfac1.1  |-  ( ph  ->  A  C_  Prime )
ablfac1c.d  |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
ablfac1.2  |-  ( ph  ->  D  C_  A )
Assertion
Ref Expression
ablfac1c  |-  ( ph  ->  ( G DProd  S )  =  B )
Distinct variable groups:    w, p, x, B    D, p, x    ph, p, w, x    A, p, x    O, p, x    G, p, x
Allowed substitution hints:    A( w)    D( w)    S( x, w, p)    G( w)    O( w)

Proof of Theorem ablfac1c
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 ablfac1.f . 2  |-  ( ph  ->  B  e.  Fin )
2 ablfac1.b . . . 4  |-  B  =  ( Base `  G
)
32dprdssv 16518 . . 3  |-  ( G DProd 
S )  C_  B
43a1i 11 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  B )
5 ssfi 7545 . . . . . 6  |-  ( ( B  e.  Fin  /\  ( G DProd  S )  C_  B )  ->  ( G DProd  S )  e.  Fin )
61, 3, 5sylancl 662 . . . . 5  |-  ( ph  ->  ( G DProd  S )  e.  Fin )
7 hashcl 12138 . . . . 5  |-  ( ( G DProd  S )  e. 
Fin  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
86, 7syl 16 . . . 4  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
9 hashcl 12138 . . . . 5  |-  ( B  e.  Fin  ->  ( # `
 B )  e. 
NN0 )
101, 9syl 16 . . . 4  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
11 ablfac1.o . . . . . . 7  |-  O  =  ( od `  G
)
12 ablfac1.s . . . . . . 7  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
13 ablfac1.g . . . . . . 7  |-  ( ph  ->  G  e.  Abel )
14 ablfac1.1 . . . . . . 7  |-  ( ph  ->  A  C_  Prime )
152, 11, 12, 13, 1, 14ablfac1b 16583 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
16 dprdsubg 16533 . . . . . 6  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
1715, 16syl 16 . . . . 5  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
182lagsubg 15755 . . . . 5  |-  ( ( ( G DProd  S )  e.  (SubGrp `  G
)  /\  B  e.  Fin )  ->  ( # `  ( G DProd  S ) )  ||  ( # `  B ) )
1917, 1, 18syl2anc 661 . . . 4  |-  ( ph  ->  ( # `  ( G DProd  S ) )  ||  ( # `  B ) )
20 breq1 4307 . . . . . . . . . . 11  |-  ( w  =  q  ->  (
w  ||  ( # `  B
)  <->  q  ||  ( # `
 B ) ) )
21 ablfac1c.d . . . . . . . . . . 11  |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
2220, 21elrab2 3131 . . . . . . . . . 10  |-  ( q  e.  D  <->  ( q  e.  Prime  /\  q  ||  ( # `  B ) ) )
23 ablfac1.2 . . . . . . . . . . 11  |-  ( ph  ->  D  C_  A )
2423sseld 3367 . . . . . . . . . 10  |-  ( ph  ->  ( q  e.  D  ->  q  e.  A ) )
2522, 24syl5bir 218 . . . . . . . . 9  |-  ( ph  ->  ( ( q  e. 
Prime  /\  q  ||  ( # `
 B ) )  ->  q  e.  A
) )
2625impl 620 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  ||  ( # `  B
) )  ->  q  e.  A )
272, 11, 12, 13, 1, 14ablfac1a 16582 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
28 fvex 5713 . . . . . . . . . . . . . . . . . . . 20  |-  ( Base `  G )  e.  _V
292, 28eqeltri 2513 . . . . . . . . . . . . . . . . . . 19  |-  B  e. 
_V
3029rabex 4455 . . . . . . . . . . . . . . . . . 18  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
3130, 12dmmpti 5552 . . . . . . . . . . . . . . . . 17  |-  dom  S  =  A
3231a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  S  =  A )
3315, 32dprdf2 16503 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
3433ffvelrnda 5855 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  G )
)
3515adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  G dom DProd  S )
3631a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  dom  S  =  A )
37 simpr 461 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  A )
3835, 36, 37dprdub 16534 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  C_  ( G DProd  S ) )
3917adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  ( G DProd  S )  e.  (SubGrp `  G ) )
40 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( Gs  ( G DProd  S ) )  =  ( Gs  ( G DProd 
S ) )
4140subsubg 15716 . . . . . . . . . . . . . . 15  |-  ( ( G DProd  S )  e.  (SubGrp `  G )  ->  ( ( S `  q )  e.  (SubGrp `  ( Gs  ( G DProd  S
) ) )  <->  ( ( S `  q )  e.  (SubGrp `  G )  /\  ( S `  q
)  C_  ( G DProd  S ) ) ) )
4239, 41syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  (
( S `  q
)  e.  (SubGrp `  ( Gs  ( G DProd  S
) ) )  <->  ( ( S `  q )  e.  (SubGrp `  G )  /\  ( S `  q
)  C_  ( G DProd  S ) ) ) )
4334, 38, 42mpbir2and 913 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  ( Gs  ( G DProd  S ) ) ) )
4440subgbas 15697 . . . . . . . . . . . . . . 15  |-  ( ( G DProd  S )  e.  (SubGrp `  G )  ->  ( G DProd  S )  =  ( Base `  ( Gs  ( G DProd  S ) ) ) )
4539, 44syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  ( G DProd  S )  =  (
Base `  ( Gs  ( G DProd  S ) ) ) )
466adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  ( G DProd  S )  e.  Fin )
4745, 46eqeltrrd 2518 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( Base `  ( Gs  ( G DProd 
S ) ) )  e.  Fin )
48 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( G DProd  S
) ) )  =  ( Base `  ( Gs  ( G DProd  S ) ) )
4948lagsubg 15755 . . . . . . . . . . . . 13  |-  ( ( ( S `  q
)  e.  (SubGrp `  ( Gs  ( G DProd  S
) ) )  /\  ( Base `  ( Gs  ( G DProd  S ) ) )  e.  Fin )  -> 
( # `  ( S `
 q ) ) 
||  ( # `  ( Base `  ( Gs  ( G DProd 
S ) ) ) ) )
5043, 47, 49syl2anc 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  ||  ( # `  ( Base `  ( Gs  ( G DProd  S
) ) ) ) )
5145fveq2d 5707 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( G DProd  S
) )  =  (
# `  ( Base `  ( Gs  ( G DProd  S
) ) ) ) )
5250, 51breqtrrd 4330 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  ||  ( # `  ( G DProd 
S ) ) )
5327, 52eqbrtrrd 4326 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
q ^ ( q 
pCnt  ( # `  B
) ) )  ||  ( # `  ( G DProd 
S ) ) )
5414sselda 3368 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  Prime )
558nn0zd 10757 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e.  ZZ )
5655adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( G DProd  S
) )  e.  ZZ )
57 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  Prime )
58 ablgrp 16294 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Abel  ->  G  e. 
Grp )
592grpbn0 15579 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  B  =/=  (/) )
6013, 58, 593syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =/=  (/) )
61 hashnncl 12146 . . . . . . . . . . . . . . . 16  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
621, 61syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
6360, 62mpbird 232 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  B
)  e.  NN )
6463adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  Prime )  ->  ( # `  B
)  e.  NN )
6557, 64pccld 13929 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  B
) )  e.  NN0 )
6654, 65syldan 470 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  e.  NN0 )
67 pcdvdsb 13947 . . . . . . . . . . 11  |-  ( ( q  e.  Prime  /\  ( # `
 ( G DProd  S
) )  e.  ZZ  /\  ( q  pCnt  ( # `
 B ) )  e.  NN0 )  -> 
( ( q  pCnt  (
# `  B )
)  <_  ( q  pCnt  ( # `  ( G DProd  S ) ) )  <-> 
( q ^ (
q  pCnt  ( # `  B
) ) )  ||  ( # `  ( G DProd 
S ) ) ) )
6854, 56, 66, 67syl3anc 1218 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
( q  pCnt  ( # `
 B ) )  <_  ( q  pCnt  (
# `  ( G DProd  S ) ) )  <->  ( q ^ ( q  pCnt  (
# `  B )
) )  ||  ( # `
 ( G DProd  S
) ) ) )
6953, 68mpbird 232 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
7069adantlr 714 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
7126, 70syldan 470 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  ||  ( # `  B
) )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
72 pceq0 13949 . . . . . . . . . 10  |-  ( ( q  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  (
( q  pCnt  ( # `
 B ) )  =  0  <->  -.  q  ||  ( # `  B
) ) )
7357, 64, 72syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  Prime )  ->  ( (
q  pCnt  ( # `  B
) )  =  0  <->  -.  q  ||  ( # `  B ) ) )
7473biimpar 485 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  -.  q  ||  ( # `  B
) )  ->  (
q  pCnt  ( # `  B
) )  =  0 )
75 eqid 2443 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  =  ( 0g `  G
)
7675subg0cl 15701 . . . . . . . . . . . . . 14  |-  ( ( G DProd  S )  e.  (SubGrp `  G )  ->  ( 0g `  G
)  e.  ( G DProd 
S ) )
77 ne0i 3655 . . . . . . . . . . . . . 14  |-  ( ( 0g `  G )  e.  ( G DProd  S
)  ->  ( G DProd  S )  =/=  (/) )
7817, 76, 773syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G DProd  S )  =/=  (/) )
79 hashnncl 12146 . . . . . . . . . . . . . 14  |-  ( ( G DProd  S )  e. 
Fin  ->  ( ( # `  ( G DProd  S ) )  e.  NN  <->  ( G DProd  S )  =/=  (/) ) )
806, 79syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( # `  ( G DProd  S ) )  e.  NN  <->  ( G DProd  S
)  =/=  (/) ) )
8178, 80mpbird 232 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e.  NN )
8281adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  Prime )  ->  ( # `  ( G DProd  S ) )  e.  NN )
8357, 82pccld 13929 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  ( G DProd  S ) ) )  e.  NN0 )
8483nn0ge0d 10651 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  Prime )  ->  0  <_  ( q  pCnt  ( # `  ( G DProd  S ) ) ) )
8584adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  -.  q  ||  ( # `  B
) )  ->  0  <_  ( q  pCnt  ( # `
 ( G DProd  S
) ) ) )
8674, 85eqbrtrd 4324 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  -.  q  ||  ( # `  B
) )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
8771, 86pm2.61dan 789 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
8887ralrimiva 2811 . . . . 5  |-  ( ph  ->  A. q  e.  Prime  ( q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
8910nn0zd 10757 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
90 pc2dvds 13957 . . . . . 6  |-  ( ( ( # `  B
)  e.  ZZ  /\  ( # `  ( G DProd 
S ) )  e.  ZZ )  ->  (
( # `  B ) 
||  ( # `  ( G DProd  S ) )  <->  A. q  e.  Prime  ( q  pCnt  (
# `  B )
)  <_  ( q  pCnt  ( # `  ( G DProd  S ) ) ) ) )
9189, 55, 90syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( # `  B
)  ||  ( # `  ( G DProd  S ) )  <->  A. q  e.  Prime  ( q  pCnt  (
# `  B )
)  <_  ( q  pCnt  ( # `  ( G DProd  S ) ) ) ) )
9288, 91mpbird 232 . . . 4  |-  ( ph  ->  ( # `  B
)  ||  ( # `  ( G DProd  S ) ) )
93 dvdseq 13592 . . . 4  |-  ( ( ( ( # `  ( G DProd  S ) )  e. 
NN0  /\  ( # `  B
)  e.  NN0 )  /\  ( ( # `  ( G DProd  S ) )  ||  ( # `  B )  /\  ( # `  B
)  ||  ( # `  ( G DProd  S ) ) ) )  ->  ( # `  ( G DProd  S ) )  =  ( # `  B
) )
948, 10, 19, 92, 93syl22anc 1219 . . 3  |-  ( ph  ->  ( # `  ( G DProd  S ) )  =  ( # `  B
) )
95 hashen 12130 . . . 4  |-  ( ( ( G DProd  S )  e.  Fin  /\  B  e.  Fin )  ->  (
( # `  ( G DProd 
S ) )  =  ( # `  B
)  <->  ( G DProd  S
)  ~~  B )
)
966, 1, 95syl2anc 661 . . 3  |-  ( ph  ->  ( ( # `  ( G DProd  S ) )  =  ( # `  B
)  <->  ( G DProd  S
)  ~~  B )
)
9794, 96mpbid 210 . 2  |-  ( ph  ->  ( G DProd  S ) 
~~  B )
98 fisseneq 7536 . 2  |-  ( ( B  e.  Fin  /\  ( G DProd  S )  C_  B  /\  ( G DProd  S
)  ~~  B )  ->  ( G DProd  S )  =  B )
991, 4, 97, 98syl3anc 1218 1  |-  ( ph  ->  ( G DProd  S )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   {crab 2731   _Vcvv 2984    C_ wss 3340   (/)c0 3649   class class class wbr 4304    e. cmpt 4362   dom cdm 4852   ` cfv 5430  (class class class)co 6103    ~~ cen 7319   Fincfn 7322   0cc0 9294    <_ cle 9431   NNcn 10334   NN0cn0 10591   ZZcz 10658   ^cexp 11877   #chash 12115    || cdivides 13547   Primecprime 13775    pCnt cpc 13915   Basecbs 14186   ↾s cress 14187   0gc0g 14390   Grpcgrp 15422  SubGrpcsubg 15687   odcod 16040   Abelcabel 16290   DProd cdprd 16487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-disj 4275  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-tpos 6757  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-omul 6937  df-er 7113  df-ec 7115  df-qs 7119  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-sup 7703  df-oi 7736  df-card 8121  df-acn 8124  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-sum 13176  df-dvds 13548  df-gcd 13703  df-prm 13776  df-pc 13916  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-0g 14392  df-gsum 14393  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mulg 15560  df-subg 15690  df-eqg 15692  df-ghm 15757  df-gim 15799  df-ga 15820  df-cntz 15847  df-oppg 15873  df-od 16044  df-lsm 16147  df-pj1 16148  df-cmn 16291  df-abl 16292  df-dprd 16489
This theorem is referenced by:  ablfaclem2  16599
  Copyright terms: Public domain W3C validator