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Theorem pgpfac1lem4 15638
Description: Lemma for pgpfac1 15640. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypotheses
Ref Expression
pgpfac1.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
pgpfac1.s  |-  S  =  ( K `  { A } )
pgpfac1.b  |-  B  =  ( Base `  G
)
pgpfac1.o  |-  O  =  ( od `  G
)
pgpfac1.e  |-  E  =  (gEx `  G )
pgpfac1.z  |-  .0.  =  ( 0g `  G )
pgpfac1.l  |-  .(+)  =  (
LSSum `  G )
pgpfac1.p  |-  ( ph  ->  P pGrp  G )
pgpfac1.g  |-  ( ph  ->  G  e.  Abel )
pgpfac1.n  |-  ( ph  ->  B  e.  Fin )
pgpfac1.oe  |-  ( ph  ->  ( O `  A
)  =  E )
pgpfac1.u  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
pgpfac1.au  |-  ( ph  ->  A  e.  U )
pgpfac1.w  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
pgpfac1.i  |-  ( ph  ->  ( S  i^i  W
)  =  {  .0.  } )
pgpfac1.ss  |-  ( ph  ->  ( S  .(+)  W ) 
C_  U )
pgpfac1.2  |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )
pgpfac1.c  |-  ( ph  ->  C  e.  ( U 
\  ( S  .(+)  W ) ) )
pgpfac1.mg  |-  .x.  =  (.g
`  G )
Assertion
Ref Expression
pgpfac1lem4  |-  ( ph  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
Distinct variable groups:    t,  .0.    w, t, A    t,  .(+) , w   
t, P, w    t, B    t, G, w    t, U, w    t, C, w   
t, S, w    t, W, w    ph, t, w   
t,  .x. , w    t, K, w
Allowed substitution hints:    B( w)    E( w, t)    O( w, t)    .0. ( w)

Proof of Theorem pgpfac1lem4
Dummy variables  k 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pgpfac1.k . . . . . . . 8  |-  K  =  (mrCls `  (SubGrp `  G
) )
2 pgpfac1.s . . . . . . . 8  |-  S  =  ( K `  { A } )
3 pgpfac1.b . . . . . . . 8  |-  B  =  ( Base `  G
)
4 pgpfac1.o . . . . . . . 8  |-  O  =  ( od `  G
)
5 pgpfac1.e . . . . . . . 8  |-  E  =  (gEx `  G )
6 pgpfac1.z . . . . . . . 8  |-  .0.  =  ( 0g `  G )
7 pgpfac1.l . . . . . . . 8  |-  .(+)  =  (
LSSum `  G )
8 pgpfac1.p . . . . . . . 8  |-  ( ph  ->  P pGrp  G )
9 pgpfac1.g . . . . . . . 8  |-  ( ph  ->  G  e.  Abel )
10 pgpfac1.n . . . . . . . 8  |-  ( ph  ->  B  e.  Fin )
11 pgpfac1.oe . . . . . . . 8  |-  ( ph  ->  ( O `  A
)  =  E )
12 pgpfac1.u . . . . . . . 8  |-  ( ph  ->  U  e.  (SubGrp `  G ) )
13 pgpfac1.au . . . . . . . 8  |-  ( ph  ->  A  e.  U )
14 pgpfac1.w . . . . . . . 8  |-  ( ph  ->  W  e.  (SubGrp `  G ) )
15 pgpfac1.i . . . . . . . 8  |-  ( ph  ->  ( S  i^i  W
)  =  {  .0.  } )
16 pgpfac1.ss . . . . . . . 8  |-  ( ph  ->  ( S  .(+)  W ) 
C_  U )
17 pgpfac1.2 . . . . . . . 8  |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )
18 pgpfac1.c . . . . . . . 8  |-  ( ph  ->  C  e.  ( U 
\  ( S  .(+)  W ) ) )
19 pgpfac1.mg . . . . . . . 8  |-  .x.  =  (.g
`  G )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19pgpfac1lem2 15635 . . . . . . 7  |-  ( ph  ->  ( P  .x.  C
)  e.  ( S 
.(+)  W ) )
21 ablgrp 15419 . . . . . . . . . . . 12  |-  ( G  e.  Abel  ->  G  e. 
Grp )
229, 21syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Grp )
233subgacs 14977 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  B ) )
24 acsmre 13879 . . . . . . . . . . 11  |-  ( (SubGrp `  G )  e.  (ACS
`  B )  -> 
(SubGrp `  G )  e.  (Moore `  B )
)
2522, 23, 243syl 19 . . . . . . . . . 10  |-  ( ph  ->  (SubGrp `  G )  e.  (Moore `  B )
)
263subgss 14947 . . . . . . . . . . . 12  |-  ( U  e.  (SubGrp `  G
)  ->  U  C_  B
)
2712, 26syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U  C_  B )
2827, 13sseldd 3351 . . . . . . . . . 10  |-  ( ph  ->  A  e.  B )
291mrcsncl 13839 . . . . . . . . . 10  |-  ( ( (SubGrp `  G )  e.  (Moore `  B )  /\  A  e.  B
)  ->  ( K `  { A } )  e.  (SubGrp `  G
) )
3025, 28, 29syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
312, 30syl5eqel 2522 . . . . . . . 8  |-  ( ph  ->  S  e.  (SubGrp `  G ) )
327lsmcom 15475 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )  /\  W  e.  (SubGrp `  G ) )  -> 
( S  .(+)  W )  =  ( W  .(+)  S ) )
339, 31, 14, 32syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( S  .(+)  W )  =  ( W  .(+)  S ) )
3420, 33eleqtrd 2514 . . . . . 6  |-  ( ph  ->  ( P  .x.  C
)  e.  ( W 
.(+)  S ) )
35 eqid 2438 . . . . . . 7  |-  ( -g `  G )  =  (
-g `  G )
3635, 7, 14, 31lsmelvalm 15287 . . . . . 6  |-  ( ph  ->  ( ( P  .x.  C )  e.  ( W  .(+)  S )  <->  E. w  e.  W  E. s  e.  S  ( P  .x.  C )  =  ( w ( -g `  G ) s ) ) )
3734, 36mpbid 203 . . . . 5  |-  ( ph  ->  E. w  e.  W  E. s  e.  S  ( P  .x.  C )  =  ( w (
-g `  G )
s ) )
38 eqid 2438 . . . . . . . . . . 11  |-  ( k  e.  ZZ  |->  ( k 
.x.  A ) )  =  ( k  e.  ZZ  |->  ( k  .x.  A ) )
393, 19, 38, 1cycsubg2 14979 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  B )  ->  ( K `  { A } )  =  ran  ( k  e.  ZZ  |->  ( k  .x.  A
) ) )
4022, 28, 39syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( K `  { A } )  =  ran  ( k  e.  ZZ  |->  ( k  .x.  A
) ) )
412, 40syl5eq 2482 . . . . . . . 8  |-  ( ph  ->  S  =  ran  (
k  e.  ZZ  |->  ( k  .x.  A ) ) )
4241rexeqdv 2913 . . . . . . 7  |-  ( ph  ->  ( E. s  e.  S  ( P  .x.  C )  =  ( w ( -g `  G
) s )  <->  E. s  e.  ran  ( k  e.  ZZ  |->  ( k  .x.  A ) ) ( P  .x.  C )  =  ( w (
-g `  G )
s ) ) )
43 ovex 6108 . . . . . . . . 9  |-  ( k 
.x.  A )  e. 
_V
4443rgenw 2775 . . . . . . . 8  |-  A. k  e.  ZZ  ( k  .x.  A )  e.  _V
45 oveq2 6091 . . . . . . . . . 10  |-  ( s  =  ( k  .x.  A )  ->  (
w ( -g `  G
) s )  =  ( w ( -g `  G ) ( k 
.x.  A ) ) )
4645eqeq2d 2449 . . . . . . . . 9  |-  ( s  =  ( k  .x.  A )  ->  (
( P  .x.  C
)  =  ( w ( -g `  G
) s )  <->  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
4738, 46rexrnmpt 5881 . . . . . . . 8  |-  ( A. k  e.  ZZ  (
k  .x.  A )  e.  _V  ->  ( E. s  e.  ran  ( k  e.  ZZ  |->  ( k 
.x.  A ) ) ( P  .x.  C
)  =  ( w ( -g `  G
) s )  <->  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
4844, 47ax-mp 8 . . . . . . 7  |-  ( E. s  e.  ran  (
k  e.  ZZ  |->  ( k  .x.  A ) ) ( P  .x.  C )  =  ( w ( -g `  G
) s )  <->  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) )
4942, 48syl6bb 254 . . . . . 6  |-  ( ph  ->  ( E. s  e.  S  ( P  .x.  C )  =  ( w ( -g `  G
) s )  <->  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
5049rexbidv 2728 . . . . 5  |-  ( ph  ->  ( E. w  e.  W  E. s  e.  S  ( P  .x.  C )  =  ( w ( -g `  G
) s )  <->  E. w  e.  W  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) ) ) )
5137, 50mpbid 203 . . . 4  |-  ( ph  ->  E. w  e.  W  E. k  e.  ZZ  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) ) )
52 rexcom 2871 . . . 4  |-  ( E. w  e.  W  E. k  e.  ZZ  ( P  .x.  C )  =  ( w ( -g `  G ) ( k 
.x.  A ) )  <->  E. k  e.  ZZ  E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) ) )
5351, 52sylib 190 . . 3  |-  ( ph  ->  E. k  e.  ZZ  E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) ) )
5422ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  G  e.  Grp )
553subgss 14947 . . . . . . . . . . 11  |-  ( W  e.  (SubGrp `  G
)  ->  W  C_  B
)
5614, 55syl 16 . . . . . . . . . 10  |-  ( ph  ->  W  C_  B )
5756adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ZZ )  ->  W  C_  B )
5857sselda 3350 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  w  e.  B )
59 simplr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  k  e.  ZZ )
6028ad2antrr 708 . . . . . . . . 9  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  A  e.  B )
613, 19mulgcl 14909 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  k  e.  ZZ  /\  A  e.  B )  ->  (
k  .x.  A )  e.  B )
6254, 59, 60, 61syl3anc 1185 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  (
k  .x.  A )  e.  B )
63 pgpprm 15229 . . . . . . . . . . 11  |-  ( P pGrp 
G  ->  P  e.  Prime )
64 prmz 13085 . . . . . . . . . . 11  |-  ( P  e.  Prime  ->  P  e.  ZZ )
658, 63, 643syl 19 . . . . . . . . . 10  |-  ( ph  ->  P  e.  ZZ )
6618eldifad 3334 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  U )
6727, 66sseldd 3351 . . . . . . . . . 10  |-  ( ph  ->  C  e.  B )
683, 19mulgcl 14909 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  P  e.  ZZ  /\  C  e.  B )  ->  ( P  .x.  C )  e.  B )
6922, 65, 67, 68syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( P  .x.  C
)  e.  B )
7069ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  ( P  .x.  C )  e.  B )
71 eqid 2438 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
723, 71, 35grpsubadd 14878 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( w  e.  B  /\  ( k  .x.  A
)  e.  B  /\  ( P  .x.  C )  e.  B ) )  ->  ( ( w ( -g `  G
) ( k  .x.  A ) )  =  ( P  .x.  C
)  <->  ( ( P 
.x.  C ) ( +g  `  G ) ( k  .x.  A
) )  =  w ) )
7354, 58, 62, 70, 72syl13anc 1187 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  (
( w ( -g `  G ) ( k 
.x.  A ) )  =  ( P  .x.  C )  <->  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) )  =  w ) )
74 eqcom 2440 . . . . . . 7  |-  ( ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) )  <->  ( w
( -g `  G ) ( k  .x.  A
) )  =  ( P  .x.  C ) )
75 eqcom 2440 . . . . . . 7  |-  ( w  =  ( ( P 
.x.  C ) ( +g  `  G ) ( k  .x.  A
) )  <->  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) )  =  w )
7673, 74, 753bitr4g 281 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  w  e.  W )  ->  (
( P  .x.  C
)  =  ( w ( -g `  G
) ( k  .x.  A ) )  <->  w  =  ( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) ) ) )
7776rexbidva 2724 . . . . 5  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) )  <->  E. w  e.  W  w  =  ( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) ) ) )
78 risset 2755 . . . . 5  |-  ( ( ( P  .x.  C
) ( +g  `  G
) ( k  .x.  A ) )  e.  W  <->  E. w  e.  W  w  =  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) ) )
7977, 78syl6bbr 256 . . . 4  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( E. w  e.  W  ( P  .x.  C )  =  ( w (
-g `  G )
( k  .x.  A
) )  <->  ( ( P  .x.  C ) ( +g  `  G ) ( k  .x.  A
) )  e.  W
) )
8079rexbidva 2724 . . 3  |-  ( ph  ->  ( E. k  e.  ZZ  E. w  e.  W  ( P  .x.  C )  =  ( w ( -g `  G
) ( k  .x.  A ) )  <->  E. k  e.  ZZ  ( ( P 
.x.  C ) ( +g  `  G ) ( k  .x.  A
) )  e.  W
) )
8153, 80mpbid 203 . 2  |-  ( ph  ->  E. k  e.  ZZ  ( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) )  e.  W )
828adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  P pGrp  G )
839adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  G  e.  Abel )
8410adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  B  e.  Fin )
8511adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( O `  A
)  =  E )
8612adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  U  e.  (SubGrp `  G
) )
8713adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  A  e.  U )
8814adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  W  e.  (SubGrp `  G
) )
8915adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( S  i^i  W
)  =  {  .0.  } )
9016adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( S  .(+)  W ) 
C_  U )
9117adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  A. w  e.  (SubGrp `  G ) ( ( w  C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )
9218adantr 453 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )
93 simprl 734 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
k  e.  ZZ )
94 simprr 735 . . 3  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  -> 
( ( P  .x.  C ) ( +g  `  G ) ( k 
.x.  A ) )  e.  W )
95 eqid 2438 . . 3  |-  ( C ( +g  `  G
) ( ( k  /  P )  .x.  A ) )  =  ( C ( +g  `  G ) ( ( k  /  P ) 
.x.  A ) )
961, 2, 3, 4, 5, 6, 7, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 19, 93, 94, 95pgpfac1lem3 15637 . 2  |-  ( (
ph  /\  ( k  e.  ZZ  /\  ( ( P  .x.  C ) ( +g  `  G
) ( k  .x.  A ) )  e.  W ) )  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
9781, 96rexlimddv 2836 1  |-  ( ph  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322    C. wpss 3323   {csn 3816   class class class wbr 4214    e. cmpt 4268   ran crn 4881   ` cfv 5456  (class class class)co 6083   Fincfn 7111    / cdiv 9679   ZZcz 10284   Primecprime 13081   Basecbs 13471   +g cplusg 13531   0gc0g 13725  Moorecmre 13809  mrClscmrc 13810  ACScacs 13812   Grpcgrp 14687   -gcsg 14690  .gcmg 14691  SubGrpcsubg 14940   odcod 15165  gExcgex 15166   pGrp cpgp 15167   LSSumclsm 15270   Abelcabel 15415
This theorem is referenced by:  pgpfac1lem5  15639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-acn 7831  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-sum 12482  df-dvds 12855  df-gcd 13009  df-prm 13082  df-pc 13213  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-subg 14943  df-eqg 14945  df-ga 15069  df-cntz 15118  df-od 15169  df-gex 15170  df-pgp 15171  df-lsm 15272  df-cmn 15416  df-abl 15417
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