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Theorem ablfac1c 15322
Description: The factors of ablfac1b 15321 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 15267 . . 3  |-  ( G DProd 
S )  C_  B
43a1i 10 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  B )
5 ssfi 7099 . . . . . 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 11366 . . . . 5  |-  ( ( G DProd  S )  e. 
Fin  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
86, 7syl 15 . . . 4  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
9 hashcl 11366 . . . . 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 15321 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
16 dprdsubg 15275 . . . . . 6  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
1715, 16syl 15 . . . . 5  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
182lagsubg 14695 . . . . 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 4042 . . . . . . . . . . 11  |-  ( w  =  q  ->  (
w  ||  ( # `  B
)  <->  q  ||  ( # `
 B ) ) )
21 ablfac1c.d . . . . . . . . . . 11  |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
2220, 21elrab2 2938 . . . . . . . . . 10  |-  ( q  e.  D  <->  ( q  e.  Prime  /\  q  ||  ( # `  B ) ) )
23 ablfac1.2 . . . . . . . . . . 11  |-  ( ph  ->  D  C_  A )
2423sseld 3192 . . . . . . . . . 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 15320 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
28 fvex 5555 . . . . . . . . . . . . . . . . . . . 20  |-  ( Base `  G )  e.  _V
292, 28eqeltri 2366 . . . . . . . . . . . . . . . . . . 19  |-  B  e. 
_V
3029rabex 4181 . . . . . . . . . . . . . . . . . 18  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
3130, 12dmmpti 5389 . . . . . . . . . . . . . . . . 17  |-  dom  S  =  A
3231a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  S  =  A )
3315, 32dprdf2 15258 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
34 ffvelrn 5679 . . . . . . . . . . . . . . 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 15276 . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . 16  |-  ( Gs  ( G DProd  S ) )  =  ( Gs  ( G DProd 
S ) )
4241subsubg 14656 . . . . . . . . . . . . . . 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 14641 . . . . . . . . . . . . . . 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 2371 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( Base `  ( Gs  ( G DProd 
S ) ) )  e.  Fin )
49 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( G DProd  S
) ) )  =  ( Base `  ( Gs  ( G DProd  S ) ) )
5049lagsubg 14695 . . . . . . . . . . . . 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 5545 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( G DProd  S
) )  =  (
# `  ( Base `  ( Gs  ( G DProd  S
) ) ) ) )
5351, 52breqtrrd 4065 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  ||  ( # `  ( G DProd 
S ) ) )
5427, 53eqbrtrrd 4061 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
q ^ ( q 
pCnt  ( # `  B
) ) )  ||  ( # `  ( G DProd 
S ) ) )
5514sselda 3193 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  Prime )
568nn0zd 10131 . . . . . . . . . . . 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 15110 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  Abel  ->  G  e. 
Grp )
6013, 59syl 15 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  Grp )
612grpbn0 14527 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  B  =/=  (/) )
6260, 61syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =/=  (/) )
63 hashnncl 11370 . . . . . . . . . . . . . . . 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 12919 . . . . . . . . . . . 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 12937 . . . . . . . . . . 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 12939 . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  G )  =  ( 0g `  G
)
7877subg0cl 14645 . . . . . . . . . . . . . . 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 3474 . . . . . . . . . . . . . 14  |-  ( ( 0g `  G )  e.  ( G DProd  S
)  ->  ( G DProd  S )  =/=  (/) )
8179, 80syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G DProd  S )  =/=  (/) )
82 hashnncl 11370 . . . . . . . . . . . . . 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 12919 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  ( G DProd  S ) ) )  e.  NN0 )
8786nn0ge0d 10037 . . . . . . . . 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 4059 . . . . . . 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 2639 . . . . 5  |-  ( ph  ->  A. q  e.  Prime  ( q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
9210nn0zd 10131 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
93 pc2dvds 12947 . . . . . 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 12592 . . . 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 11362 . . . 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 7090 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    ~~ cen 6876   Fincfn 6879   0cc0 8753    <_ cle 8884   NNcn 9762   NN0cn0 9981   ZZcz 10040   ^cexp 11120   #chash 11353    || cdivides 12547   Primecprime 12774    pCnt cpc 12905   Basecbs 13164   ↾s cress 13165   0gc0g 13416   Grpcgrp 14378  SubGrpcsubg 14631   odcod 14856   Abelcabel 15106   DProd cdprd 15247
This theorem is referenced by:  ablfaclem2  15337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-eqg 14636  df-ghm 14697  df-gim 14739  df-ga 14760  df-cntz 14809  df-oppg 14835  df-od 14860  df-lsm 14963  df-pj1 14964  df-cmn 15107  df-abl 15108  df-dprd 15249
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