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Theorem ablfac1c 16936
Description: The factors of ablfac1b 16935 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 16870 . . 3  |-  ( G DProd 
S )  C_  B
43a1i 11 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  B )
5 ssfi 7741 . . . . . 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 12397 . . . . 5  |-  ( ( G DProd  S )  e. 
Fin  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
86, 7syl 16 . . . 4  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
9 hashcl 12397 . . . . 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 16935 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
16 dprdsubg 16885 . . . . . 6  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
1715, 16syl 16 . . . . 5  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
182lagsubg 16077 . . . . 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 4450 . . . . . . . . . . 11  |-  ( w  =  q  ->  (
w  ||  ( # `  B
)  <->  q  ||  ( # `
 B ) ) )
21 ablfac1c.d . . . . . . . . . . 11  |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
2220, 21elrab2 3263 . . . . . . . . . 10  |-  ( q  e.  D  <->  ( q  e.  Prime  /\  q  ||  ( # `  B ) ) )
23 ablfac1.2 . . . . . . . . . . 11  |-  ( ph  ->  D  C_  A )
2423sseld 3503 . . . . . . . . . 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 16934 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
28 fvex 5876 . . . . . . . . . . . . . . . . . . . 20  |-  ( Base `  G )  e.  _V
292, 28eqeltri 2551 . . . . . . . . . . . . . . . . . . 19  |-  B  e. 
_V
3029rabex 4598 . . . . . . . . . . . . . . . . . 18  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
3130, 12dmmpti 5710 . . . . . . . . . . . . . . . . 17  |-  dom  S  =  A
3231a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  S  =  A )
3315, 32dprdf2 16855 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
3433ffvelrnda 6022 . . . . . . . . . . . . . 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 16886 . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . 16  |-  ( Gs  ( G DProd  S ) )  =  ( Gs  ( G DProd 
S ) )
4140subsubg 16038 . . . . . . . . . . . . . . 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 920 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  ( Gs  ( G DProd  S ) ) ) )
4440subgbas 16019 . . . . . . . . . . . . . . 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 2556 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( Base `  ( Gs  ( G DProd 
S ) ) )  e.  Fin )
48 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( G DProd  S
) ) )  =  ( Base `  ( Gs  ( G DProd  S ) ) )
4948lagsubg 16077 . . . . . . . . . . . . 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 5870 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( G DProd  S
) )  =  (
# `  ( Base `  ( Gs  ( G DProd  S
) ) ) ) )
5250, 51breqtrrd 4473 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  ||  ( # `  ( G DProd 
S ) ) )
5327, 52eqbrtrrd 4469 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
q ^ ( q 
pCnt  ( # `  B
) ) )  ||  ( # `  ( G DProd 
S ) ) )
5414sselda 3504 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  Prime )
558nn0zd 10965 . . . . . . . . . . . 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 16618 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Abel  ->  G  e. 
Grp )
592grpbn0 15893 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  B  =/=  (/) )
6013, 58, 593syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =/=  (/) )
61 hashnncl 12405 . . . . . . . . . . . . . . . 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 14236 . . . . . . . . . . . 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 14254 . . . . . . . . . . 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 1228 . . . . . . . . . 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 14256 . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  =  ( 0g `  G
)
7675subg0cl 16023 . . . . . . . . . . . . . 14  |-  ( ( G DProd  S )  e.  (SubGrp `  G )  ->  ( 0g `  G
)  e.  ( G DProd 
S ) )
77 ne0i 3791 . . . . . . . . . . . . . 14  |-  ( ( 0g `  G )  e.  ( G DProd  S
)  ->  ( G DProd  S )  =/=  (/) )
7817, 76, 773syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G DProd  S )  =/=  (/) )
79 hashnncl 12405 . . . . . . . . . . . . . 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 14236 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  ( G DProd  S ) ) )  e.  NN0 )
8483nn0ge0d 10856 . . . . . . . . 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 4467 . . . . . . 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 2878 . . . . 5  |-  ( ph  ->  A. q  e.  Prime  ( q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
8910nn0zd 10965 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
90 pc2dvds 14264 . . . . . 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 13895 . . . 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 1229 . . 3  |-  ( ph  ->  ( # `  ( G DProd  S ) )  =  ( # `  B
) )
95 hashen 12389 . . . 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 7732 . 2  |-  ( ( B  e.  Fin  /\  ( G DProd  S )  C_  B  /\  ( G DProd  S
)  ~~  B )  ->  ( G DProd  S )  =  B )
991, 4, 97, 98syl3anc 1228 1  |-  ( ph  ->  ( G DProd  S )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ` cfv 5588  (class class class)co 6285    ~~ cen 7514   Fincfn 7517   0cc0 9493    <_ cle 9630   NNcn 10537   NN0cn0 10796   ZZcz 10865   ^cexp 12135   #chash 12374    || cdivides 13850   Primecprime 14079    pCnt cpc 14222   Basecbs 14493   ↾s cress 14494   0gc0g 14698   Grpcgrp 15730  SubGrpcsubg 16009   odcod 16364   Abelcabl 16614   DProd cdprd 16839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-tpos 6956  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-omul 7136  df-er 7312  df-ec 7314  df-qs 7318  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-sup 7902  df-oi 7936  df-card 8321  df-acn 8324  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-q 11184  df-rp 11222  df-fz 11674  df-fzo 11794  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-fac 12323  df-bc 12350  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277  df-sum 13475  df-dvds 13851  df-gcd 14007  df-prm 14080  df-pc 14223  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-0g 14700  df-gsum 14701  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-mhm 15789  df-submnd 15790  df-grp 15871  df-minusg 15872  df-sbg 15873  df-mulg 15874  df-subg 16012  df-eqg 16014  df-ghm 16079  df-gim 16121  df-ga 16142  df-cntz 16169  df-oppg 16195  df-od 16368  df-lsm 16471  df-pj1 16472  df-cmn 16615  df-abl 16616  df-dprd 16841
This theorem is referenced by:  ablfaclem2  16951
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