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Theorem ablfac1c 17249
Description: The factors of ablfac1b 17248 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 17183 . . 3  |-  ( G DProd 
S )  C_  B
43a1i 11 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  B )
5 ssfi 7759 . . . . . 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 12431 . . . . 5  |-  ( ( G DProd  S )  e. 
Fin  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
86, 7syl 16 . . . 4  |-  ( ph  ->  ( # `  ( G DProd  S ) )  e. 
NN0 )
9 hashcl 12431 . . . . 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 17248 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
16 dprdsubg 17198 . . . . . 6  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
1715, 16syl 16 . . . . 5  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
182lagsubg 16390 . . . . 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 4459 . . . . . . . . . . 11  |-  ( w  =  q  ->  (
w  ||  ( # `  B
)  <->  q  ||  ( # `
 B ) ) )
21 ablfac1c.d . . . . . . . . . . 11  |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }
2220, 21elrab2 3259 . . . . . . . . . 10  |-  ( q  e.  D  <->  ( q  e.  Prime  /\  q  ||  ( # `  B ) ) )
23 ablfac1.2 . . . . . . . . . . 11  |-  ( ph  ->  D  C_  A )
2423sseld 3498 . . . . . . . . . 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 17247 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  =  ( q ^ (
q  pCnt  ( # `  B
) ) ) )
28 fvex 5882 . . . . . . . . . . . . . . . . . . . 20  |-  ( Base `  G )  e.  _V
292, 28eqeltri 2541 . . . . . . . . . . . . . . . . . . 19  |-  B  e. 
_V
3029rabex 4607 . . . . . . . . . . . . . . . . . 18  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
3130, 12dmmpti 5716 . . . . . . . . . . . . . . . . 17  |-  dom  S  =  A
3231a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  S  =  A )
3315, 32dprdf2 17167 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S : A --> (SubGrp `  G ) )
3433ffvelrnda 6032 . . . . . . . . . . . . . 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 17199 . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . 16  |-  ( Gs  ( G DProd  S ) )  =  ( Gs  ( G DProd 
S ) )
4140subsubg 16351 . . . . . . . . . . . . . . 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 922 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( S `  q )  e.  (SubGrp `  ( Gs  ( G DProd  S ) ) ) )
4440subgbas 16332 . . . . . . . . . . . . . . 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 2546 . . . . . . . . . . . . 13  |-  ( (
ph  /\  q  e.  A )  ->  ( Base `  ( Gs  ( G DProd 
S ) ) )  e.  Fin )
48 eqid 2457 . . . . . . . . . . . . . 14  |-  ( Base `  ( Gs  ( G DProd  S
) ) )  =  ( Base `  ( Gs  ( G DProd  S ) ) )
4948lagsubg 16390 . . . . . . . . . . . . 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 5876 . . . . . . . . . . . 12  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( G DProd  S
) )  =  (
# `  ( Base `  ( Gs  ( G DProd  S
) ) ) ) )
5250, 51breqtrrd 4482 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  ( # `
 ( S `  q ) )  ||  ( # `  ( G DProd 
S ) ) )
5327, 52eqbrtrrd 4478 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  A )  ->  (
q ^ ( q 
pCnt  ( # `  B
) ) )  ||  ( # `  ( G DProd 
S ) ) )
5414sselda 3499 . . . . . . . . . . 11  |-  ( (
ph  /\  q  e.  A )  ->  q  e.  Prime )
558nn0zd 10988 . . . . . . . . . . . 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 16930 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Abel  ->  G  e. 
Grp )
592grpbn0 16206 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  B  =/=  (/) )
6013, 58, 593syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =/=  (/) )
61 hashnncl 12439 . . . . . . . . . . . . . . . 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 14386 . . . . . . . . . . . 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 14404 . . . . . . . . . . 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 14406 . . . . . . . . . 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 2457 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  =  ( 0g `  G
)
7675subg0cl 16336 . . . . . . . . . . . . . 14  |-  ( ( G DProd  S )  e.  (SubGrp `  G )  ->  ( 0g `  G
)  e.  ( G DProd 
S ) )
77 ne0i 3799 . . . . . . . . . . . . . 14  |-  ( ( 0g `  G )  e.  ( G DProd  S
)  ->  ( G DProd  S )  =/=  (/) )
7817, 76, 773syl 20 . . . . . . . . . . . . 13  |-  ( ph  ->  ( G DProd  S )  =/=  (/) )
79 hashnncl 12439 . . . . . . . . . . . . . 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 14386 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  ( G DProd  S ) ) )  e.  NN0 )
8483nn0ge0d 10876 . . . . . . . . 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 4476 . . . . . . 7  |-  ( ( ( ph  /\  q  e.  Prime )  /\  -.  q  ||  ( # `  B
) )  ->  (
q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
8771, 86pm2.61dan 791 . . . . . 6  |-  ( (
ph  /\  q  e.  Prime )  ->  ( q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
8887ralrimiva 2871 . . . . 5  |-  ( ph  ->  A. q  e.  Prime  ( q  pCnt  ( # `  B
) )  <_  (
q  pCnt  ( # `  ( G DProd  S ) ) ) )
8910nn0zd 10988 . . . . . 6  |-  ( ph  ->  ( # `  B
)  e.  ZZ )
90 pc2dvds 14414 . . . . . 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 14045 . . . 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 12423 . . . 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 7750 . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008   ` cfv 5594  (class class class)co 6296    ~~ cen 7532   Fincfn 7535   0cc0 9509    <_ cle 9646   NNcn 10556   NN0cn0 10816   ZZcz 10885   ^cexp 12169   #chash 12408    || cdvds 13998   Primecprime 14229    pCnt cpc 14372   Basecbs 14644   ↾s cress 14645   0gc0g 14857   Grpcgrp 16180  SubGrpcsubg 16322   odcod 16676   Abelcabl 16926   DProd cdprd 17151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-sum 13521  df-dvds 13999  df-gcd 14157  df-prm 14230  df-pc 14373  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-0g 14859  df-gsum 14860  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-eqg 16327  df-ghm 16392  df-gim 16434  df-ga 16455  df-cntz 16482  df-oppg 16508  df-od 16680  df-lsm 16783  df-pj1 16784  df-cmn 16927  df-abl 16928  df-dprd 17153
This theorem is referenced by:  ablfaclem2  17264
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