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Theorem ablfac1c 15306
Description: The factors of ablfac1b 15305 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 15251 . . 3  |-  ( G DProd 
S )  C_  B
43a1i 10 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  B )
5 ssfi 7083 . . . . . 6  |-  ( ( B  e.  Fin  /\  ( G DProd  S )  C_  B )  ->  ( G DProd  S )  e.  Fin )
61, 3, 5sylancl 643 . . . . 5  |-  ( ph  ->  ( G DProd  S )  e.  Fin )
7 hashcl 11350 . . . . 5  |-  ( ( G DProd  S )  e. 
Fin  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
86, 7syl 15 . . . 4  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
9 hashcl 11350 . . . . 5  |-  ( B  e.  Fin  ->  ( # `
 B )  e. 
NN0 )
101, 9syl 15 . . . 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 15305 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
16 dprdsubg 15259 . . . . . 6  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
1715, 16syl 15 . . . . 5  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
182lagsubg 14679 . . . . 5  |-  ( ( ( G DProd  S )  e.  (SubGrp `  G
)  /\  B  e.  Fin )  ->  ( # `  ( G DProd  S ) )  ||  ( # `  B ) )
1917, 1, 18syl2anc 642 . . . 4  |-  ( ph  ->  ( # `  ( G DProd  S ) )  ||  ( # `  B ) )
20 breq1 4026 . . . . . . . . . . 11  |-  ( w  =  q  ->  (
w  ||  ( # `  B
)  <->  q  ||  ( # `
 B ) ) )
21 ablfac1c.d . . . . . . . . . . 11  |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
2220, 21elrab2 2925 . . . . . . . . . 10  |-  ( q  e.  D  <->  ( q  e.  Prime  /\  q  ||  ( # `  B ) ) )
23 ablfac1.2 . . . . . . . . . . 11  |-  ( ph  ->  D  C_  A )
2423sseld 3179 . . . . . . . . . 10  |-  ( ph  ->  ( q  e.  D  ->  q  e.  A ) )
2522, 24syl5bir 209 . . . . . . . . 9  |-  ( ph  ->  ( ( q  e. 
Prime  /\  q  ||  ( # `
 B ) )  ->  q  e.  A
) )
2625impl 603 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  ||  ( # `  B
) )  ->  q  e.  A )
272, 11, 12, 13, 1, 14ablfac1a 15304 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
28 fvex 5539 . . . . . . . . . . . . . . . . . . . 20  |-  ( Base `  G )  e.  _V
292, 28eqeltri 2353 . . . . . . . . . . . . . . . . . . 19  |-  B  e. 
_V
3029rabex 4165 . . . . . . . . . . . . . . . . . 18  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
3130, 12dmmpti 5373 . . . . . . . . . . . . . . . . 17  |-  dom  S  =  A
3231a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  S  =  A )
3315, 32dprdf2 15242 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
34 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( S : A --> (SubGrp `  G )  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  G )
)
3533, 34sylan 457 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  G )
)
3615adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  G dom DProd  S )
3731a1i 10 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  dom  S  =  A )
38 simpr 447 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  A )
3936, 37, 38dprdub 15260 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  C_  ( G DProd  S ) )
4017adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  q  e.  A )  ->  ( G DProd  S )  e.  (SubGrp `  G ) )
41 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( Gs  ( G DProd  S ) )  =  ( Gs  ( G DProd 
S ) )
4241subsubg 14640 . . . . . . . . . . . . . . 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 ) ) ) )
4340, 42syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  (
( S `  q
)  e.  (SubGrp `  ( Gs  ( G DProd  S
) ) )  <->  ( ( S `  q )  e.  (SubGrp `  G )  /\  ( S `  q
)  C_  ( G DProd  S ) ) ) )
4435, 39, 43mpbir2and 888 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  ( Gs  ( G DProd  S ) ) ) )
4541subgbas 14625 . . . . . . . . . . . . . . 15  |-  ( ( G DProd  S )  e.  (SubGrp `  G )  ->  ( G DProd  S )  =  ( Base `  ( Gs  ( G DProd  S ) ) ) )
4640, 45syl 15 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  ( G DProd  S )  =  (
Base `  ( Gs  ( G DProd  S ) ) ) )
476adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  q  e.  A )  ->  ( G DProd  S )  e.  Fin )
4846, 47eqeltrrd 2358 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( Base `  ( Gs  ( G DProd 
S ) ) )  e.  Fin )
49 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( G DProd  S
) ) )  =  ( Base `  ( Gs  ( G DProd  S ) ) )
5049lagsubg 14679 . . . . . . . . . . . . 13  |-  ( ( ( S `  q
)  e.  (SubGrp `  ( Gs  ( G DProd  S
) ) )  /\  ( Base `  ( Gs  ( G DProd  S ) ) )  e.  Fin )  -> 
( # `  ( S `
 q ) ) 
||  ( # `  ( Base `  ( Gs  ( G DProd 
S ) ) ) ) )
5144, 48, 50syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  ||  ( # `  ( Base `  ( Gs  ( G DProd  S
) ) ) ) )
5246fveq2d 5529 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( G DProd  S
) )  =  (
# `  ( Base `  ( Gs  ( G DProd  S
) ) ) ) )
5351, 52breqtrrd 4049 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  ||  ( # `  ( G DProd 
S ) ) )
5427, 53eqbrtrrd 4045 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
q ^ ( q 
pCnt  ( # `  B
) ) )  ||  ( # `  ( G DProd 
S ) ) )
5514sselda 3180 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  Prime )
568nn0zd 10115 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e.  ZZ )
5756adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( G DProd  S
) )  e.  ZZ )
58 simpr 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  Prime )  ->  q  e.  Prime )
59 ablgrp 15094 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  Abel  ->  G  e. 
Grp )
6013, 59syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  Grp )
612grpbn0 14511 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  B  =/=  (/) )
6260, 61syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =/=  (/) )
63 hashnncl 11354 . . . . . . . . . . . . . . . 16  |-  ( B  e.  Fin  ->  (
( # `  B )  e.  NN  <->  B  =/=  (/) ) )
641, 63syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( # `  B
)  e.  NN  <->  B  =/=  (/) ) )
6562, 64mpbird 223 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( # `  B
)  e.  NN )
6665adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  Prime )  ->  ( # `  B
)  e.  NN )
6758, 66pccld 12903 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  B
) )  e.  NN0 )
6855, 67syldan 456 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  e.  NN0 )
69 pcdvdsb 12921 . . . . . . . . . . 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 ) ) ) )
7055, 57, 68, 69syl3anc 1182 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
( q  pCnt  ( # `
 B ) )  <_  ( q  pCnt  (
# `  ( G DProd  S ) ) )  <->  ( q ^ ( q  pCnt  (
# `  B )
) )  ||  ( # `
 ( G DProd  S
) ) ) )
7154, 70mpbird 223 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
7271adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  e.  A )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
7326, 72syldan 456 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  q  ||  ( # `  B
) )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
74 pceq0 12923 . . . . . . . . . 10  |-  ( ( q  e.  Prime  /\  ( # `
 B )  e.  NN )  ->  (
( q  pCnt  ( # `
 B ) )  =  0  <->  -.  q  ||  ( # `  B
) ) )
7558, 66, 74syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  Prime )  ->  ( (
q  pCnt  ( # `  B
) )  =  0  <->  -.  q  ||  ( # `  B ) ) )
7675biimpar 471 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  -.  q  ||  ( # `  B
) )  ->  (
q  pCnt  ( # `  B
) )  =  0 )
77 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  G )  =  ( 0g `  G
)
7877subg0cl 14629 . . . . . . . . . . . . . . 15  |-  ( ( G DProd  S )  e.  (SubGrp `  G )  ->  ( 0g `  G
)  e.  ( G DProd 
S ) )
7917, 78syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 0g `  G
)  e.  ( G DProd 
S ) )
80 ne0i 3461 . . . . . . . . . . . . . 14  |-  ( ( 0g `  G )  e.  ( G DProd  S
)  ->  ( G DProd  S )  =/=  (/) )
8179, 80syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G DProd  S )  =/=  (/) )
82 hashnncl 11354 . . . . . . . . . . . . . 14  |-  ( ( G DProd  S )  e. 
Fin  ->  ( ( # `  ( G DProd  S ) )  e.  NN  <->  ( G DProd  S )  =/=  (/) ) )
836, 82syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( # `  ( G DProd  S ) )  e.  NN  <->  ( G DProd  S
)  =/=  (/) ) )
8481, 83mpbird 223 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e.  NN )
8584adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  Prime )  ->  ( # `  ( G DProd  S ) )  e.  NN )
8658, 85pccld 12903 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  ( G DProd  S ) ) )  e.  NN0 )
8786nn0ge0d 10021 . . . . . . . . 9  |-  ( (
ph  /\  q  e.  Prime )  ->  0  <_  ( q  pCnt  ( # `  ( G DProd  S ) ) ) )
8887adantr 451 . . . . . . . 8  |-  ( ( ( ph  /\  q  e.  Prime )  /\  -.  q  ||  ( # `  B
) )  ->  0  <_  ( q  pCnt  ( # `
 ( G DProd  S
) ) ) )
8976, 88eqbrtrd 4043 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  -.  q  ||  ( # `  B
) )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
9073, 89pm2.61dan 766 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
9190ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. q  e.  Prime  ( q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
9210nn0zd 10115 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
93 pc2dvds 12931 . . . . . 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 ) ) ) ) )
9492, 56, 93syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( # `  B
)  ||  ( # `  ( G DProd  S ) )  <->  A. q  e.  Prime  ( q  pCnt  (
# `  B )
)  <_  ( q  pCnt  ( # `  ( G DProd  S ) ) ) ) )
9591, 94mpbird 223 . . . 4  |-  ( ph  ->  ( # `  B
)  ||  ( # `  ( G DProd  S ) ) )
96 dvdseq 12576 . . . 4  |-  ( ( ( ( # `  ( G DProd  S ) )  e. 
NN0  /\  ( # `  B
)  e.  NN0 )  /\  ( ( # `  ( G DProd  S ) )  ||  ( # `  B )  /\  ( # `  B
)  ||  ( # `  ( G DProd  S ) ) ) )  ->  ( # `  ( G DProd  S ) )  =  ( # `  B
) )
978, 10, 19, 95, 96syl22anc 1183 . . 3  |-  ( ph  ->  ( # `  ( G DProd  S ) )  =  ( # `  B
) )
98 hashen 11346 . . . 4  |-  ( ( ( G DProd  S )  e.  Fin  /\  B  e.  Fin )  ->  (
( # `  ( G DProd 
S ) )  =  ( # `  B
)  <->  ( G DProd  S
)  ~~  B )
)
996, 1, 98syl2anc 642 . . 3  |-  ( ph  ->  ( ( # `  ( G DProd  S ) )  =  ( # `  B
)  <->  ( G DProd  S
)  ~~  B )
)
10097, 99mpbid 201 . 2  |-  ( ph  ->  ( G DProd  S ) 
~~  B )
101 fisseneq 7074 . 2  |-  ( ( B  e.  Fin  /\  ( G DProd  S )  C_  B  /\  ( G DProd  S
)  ~~  B )  ->  ( G DProd  S )  =  B )
1021, 4, 100, 101syl3anc 1182 1  |-  ( ph  ->  ( G DProd  S )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   Fincfn 6863   0cc0 8737    <_ cle 8868   NNcn 9746   NN0cn0 9965   ZZcz 10024   ^cexp 11104   #chash 11337    || cdivides 12531   Primecprime 12758    pCnt cpc 12889   Basecbs 13148   ↾s cress 13149   0gc0g 13400   Grpcgrp 14362  SubGrpcsubg 14615   odcod 14840   Abelcabel 15090   DProd cdprd 15231
This theorem is referenced by:  ablfaclem2  15321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-ghm 14681  df-gim 14723  df-ga 14744  df-cntz 14793  df-oppg 14819  df-od 14844  df-lsm 14947  df-pj1 14948  df-cmn 15091  df-abl 15092  df-dprd 15233
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