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Theorem sylow3lem3 15263
Description: Lemma for sylow3 15267, first part. The number of Sylow subgroups is the same as the index (number of cosets) of the normalizer of the Sylow subgroup  K. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypotheses
Ref Expression
sylow3.x  |-  X  =  ( Base `  G
)
sylow3.g  |-  ( ph  ->  G  e.  Grp )
sylow3.xf  |-  ( ph  ->  X  e.  Fin )
sylow3.p  |-  ( ph  ->  P  e.  Prime )
sylow3lem1.a  |-  .+  =  ( +g  `  G )
sylow3lem1.d  |-  .-  =  ( -g `  G )
sylow3lem1.m  |-  .(+)  =  ( x  e.  X , 
y  e.  ( P pSyl 
G )  |->  ran  (
z  e.  y  |->  ( ( x  .+  z
)  .-  x )
) )
sylow3lem2.k  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
sylow3lem2.h  |-  H  =  { u  e.  X  |  ( u  .(+)  K )  =  K }
sylow3lem2.n  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  K  <->  ( y  .+  x )  e.  K ) }
Assertion
Ref Expression
sylow3lem3  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) )
Distinct variable groups:    x, u, y, z,  .-    u,  .(+) , x, y, z    x, H, y    u, K, x, y, z    u, N, z    u, X, x, y, z    u, G, x, y, z    ph, u, x, y, z    u,  .+ , x, y, z    u, P, x, y, z
Allowed substitution hints:    H( z, u)    N( x, y)

Proof of Theorem sylow3lem3
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow3.xf . . . . . 6  |-  ( ph  ->  X  e.  Fin )
2 pwfi 7402 . . . . . 6  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
31, 2sylib 189 . . . . 5  |-  ( ph  ->  ~P X  e.  Fin )
4 slwsubg 15244 . . . . . . . 8  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  (SubGrp `  G ) )
5 sylow3.x . . . . . . . . 9  |-  X  =  ( Base `  G
)
65subgss 14945 . . . . . . . 8  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
74, 6syl 16 . . . . . . 7  |-  ( x  e.  ( P pSyl  G
)  ->  x  C_  X
)
8 vex 2959 . . . . . . . 8  |-  x  e. 
_V
98elpw 3805 . . . . . . 7  |-  ( x  e.  ~P X  <->  x  C_  X
)
107, 9sylibr 204 . . . . . 6  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  ~P X )
1110ssriv 3352 . . . . 5  |-  ( P pSyl 
G )  C_  ~P X
12 ssfi 7329 . . . . 5  |-  ( ( ~P X  e.  Fin  /\  ( P pSyl  G ) 
C_  ~P X )  -> 
( P pSyl  G )  e.  Fin )
133, 11, 12sylancl 644 . . . 4  |-  ( ph  ->  ( P pSyl  G )  e.  Fin )
14 hashcl 11639 . . . 4  |-  ( ( P pSyl  G )  e. 
Fin  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
1513, 14syl 16 . . 3  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
1615nn0cnd 10276 . 2  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e.  CC )
17 sylow3.g . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
18 sylow3lem2.n . . . . . . . . 9  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  K  <->  ( y  .+  x )  e.  K ) }
19 sylow3lem1.a . . . . . . . . 9  |-  .+  =  ( +g  `  G )
2018, 5, 19nmzsubg 14981 . . . . . . . 8  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
2117, 20syl 16 . . . . . . 7  |-  ( ph  ->  N  e.  (SubGrp `  G ) )
22 eqid 2436 . . . . . . . 8  |-  ( G ~QG  N )  =  ( G ~QG  N )
235, 22eqger 14990 . . . . . . 7  |-  ( N  e.  (SubGrp `  G
)  ->  ( G ~QG  N
)  Er  X )
2421, 23syl 16 . . . . . 6  |-  ( ph  ->  ( G ~QG  N )  Er  X
)
2524qsss 6965 . . . . 5  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  C_  ~P X )
26 ssfi 7329 . . . . 5  |-  ( ( ~P X  e.  Fin  /\  ( X /. ( G ~QG  N ) )  C_  ~P X )  ->  ( X /. ( G ~QG  N ) )  e.  Fin )
273, 25, 26syl2anc 643 . . . 4  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  e. 
Fin )
28 hashcl 11639 . . . 4  |-  ( ( X /. ( G ~QG  N ) )  e.  Fin  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
2927, 28syl 16 . . 3  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
3029nn0cnd 10276 . 2  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  CC )
31 eqid 2436 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
3231subg0cl 14952 . . . . . 6  |-  ( N  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  N
)
3321, 32syl 16 . . . . 5  |-  ( ph  ->  ( 0g `  G
)  e.  N )
34 ne0i 3634 . . . . 5  |-  ( ( 0g `  G )  e.  N  ->  N  =/=  (/) )
3533, 34syl 16 . . . 4  |-  ( ph  ->  N  =/=  (/) )
365subgss 14945 . . . . . . 7  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
3721, 36syl 16 . . . . . 6  |-  ( ph  ->  N  C_  X )
38 ssfi 7329 . . . . . 6  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
391, 37, 38syl2anc 643 . . . . 5  |-  ( ph  ->  N  e.  Fin )
40 hashnncl 11645 . . . . 5  |-  ( N  e.  Fin  ->  (
( # `  N )  e.  NN  <->  N  =/=  (/) ) )
4139, 40syl 16 . . . 4  |-  ( ph  ->  ( ( # `  N
)  e.  NN  <->  N  =/=  (/) ) )
4235, 41mpbird 224 . . 3  |-  ( ph  ->  ( # `  N
)  e.  NN )
4342nncnd 10016 . 2  |-  ( ph  ->  ( # `  N
)  e.  CC )
4442nnne0d 10044 . 2  |-  ( ph  ->  ( # `  N
)  =/=  0 )
45 sylow3.p . . . . 5  |-  ( ph  ->  P  e.  Prime )
46 sylow3lem1.d . . . . 5  |-  .-  =  ( -g `  G )
47 sylow3lem1.m . . . . 5  |-  .(+)  =  ( x  e.  X , 
y  e.  ( P pSyl 
G )  |->  ran  (
z  e.  y  |->  ( ( x  .+  z
)  .-  x )
) )
485, 17, 1, 45, 19, 46, 47sylow3lem1 15261 . . . 4  |-  ( ph  -> 
.(+)  e.  ( G  GrpAct  ( P pSyl  G )
) )
49 sylow3lem2.k . . . 4  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
50 sylow3lem2.h . . . . 5  |-  H  =  { u  e.  X  |  ( u  .(+)  K )  =  K }
51 eqid 2436 . . . . 5  |-  ( G ~QG  H )  =  ( G ~QG  H )
52 eqid 2436 . . . . 5  |-  { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  =  { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }
535, 50, 51, 52orbsta2 15091 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  ( P pSyl  G
) )  /\  K  e.  ( P pSyl  G ) )  /\  X  e. 
Fin )  ->  ( # `
 X )  =  ( ( # `  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )  x.  ( # `  H
) ) )
5448, 49, 1, 53syl21anc 1183 . . 3  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  [ K ] { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )  x.  ( # `  H
) ) )
555, 22, 21, 1lagsubg2 15001 . . 3  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
5652, 5gaorber 15085 . . . . . . . 8  |-  (  .(+)  e.  ( G  GrpAct  ( P pSyl 
G ) )  ->  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  Er  ( P pSyl  G ) )
5748, 56syl 16 . . . . . . 7  |-  ( ph  ->  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  Er  ( P pSyl  G ) )
5857ecss 6946 . . . . . 6  |-  ( ph  ->  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } 
C_  ( P pSyl  G
) )
5949adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  K  e.  ( P pSyl  G )
)
60 simpr 448 . . . . . . . . . 10  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  h  e.  ( P pSyl  G )
)
611adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  X  e.  Fin )
625, 61, 60, 59, 19, 46sylow2 15260 . . . . . . . . . . 11  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  E. u  e.  X  h  =  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) ) )
63 eqcom 2438 . . . . . . . . . . . . 13  |-  ( ( u  .(+)  K )  =  h  <->  h  =  (
u  .(+)  K ) )
64 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  u  e.  X )
6559adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  K  e.  ( P pSyl  G ) )
66 mptexg 5965 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  ( P pSyl  G
)  ->  ( z  e.  K  |->  ( ( u  .+  z ) 
.-  u ) )  e.  _V )
6765, 66syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  (
z  e.  K  |->  ( ( u  .+  z
)  .-  u )
)  e.  _V )
68 rnexg 5131 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  K  |->  ( ( u  .+  z
)  .-  u )
)  e.  _V  ->  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) )  e.  _V )
6967, 68syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) )  e.  _V )
70 simpr 448 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  u  /\  y  =  K )  ->  y  =  K )
71 simpl 444 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  u  /\  y  =  K )  ->  x  =  u )
7271oveq1d 6096 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  u  /\  y  =  K )  ->  ( x  .+  z
)  =  ( u 
.+  z ) )
7372, 71oveq12d 6099 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  u  /\  y  =  K )  ->  ( ( x  .+  z )  .-  x
)  =  ( ( u  .+  z ) 
.-  u ) )
7470, 73mpteq12dv 4287 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  u  /\  y  =  K )  ->  ( z  e.  y 
|->  ( ( x  .+  z )  .-  x
) )  =  ( z  e.  K  |->  ( ( u  .+  z
)  .-  u )
) )
7574rneqd 5097 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  u  /\  y  =  K )  ->  ran  ( z  e.  y  |->  ( ( x 
.+  z )  .-  x ) )  =  ran  ( z  e.  K  |->  ( ( u 
.+  z )  .-  u ) ) )
7675, 47ovmpt2ga 6203 . . . . . . . . . . . . . . 15  |-  ( ( u  e.  X  /\  K  e.  ( P pSyl  G )  /\  ran  (
z  e.  K  |->  ( ( u  .+  z
)  .-  u )
)  e.  _V )  ->  ( u  .(+)  K )  =  ran  ( z  e.  K  |->  ( ( u  .+  z ) 
.-  u ) ) )
7764, 65, 69, 76syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  (
u  .(+)  K )  =  ran  ( z  e.  K  |->  ( ( u 
.+  z )  .-  u ) ) )
7877eqeq2d 2447 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  (
h  =  ( u 
.(+)  K )  <->  h  =  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) ) ) )
7963, 78syl5bb 249 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  (
( u  .(+)  K )  =  h  <->  h  =  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) ) ) )
8079rexbidva 2722 . . . . . . . . . . 11  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  ( E. u  e.  X  (
u  .(+)  K )  =  h  <->  E. u  e.  X  h  =  ran  ( z  e.  K  |->  ( ( u  .+  z ) 
.-  u ) ) ) )
8162, 80mpbird 224 . . . . . . . . . 10  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  E. u  e.  X  ( u  .(+) 
K )  =  h )
8252gaorb 15084 . . . . . . . . . 10  |-  ( K { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } h  <->  ( K  e.  ( P pSyl  G )  /\  h  e.  ( P pSyl  G )  /\  E. u  e.  X  ( u  .(+)  K )  =  h ) )
8359, 60, 81, 82syl3anbrc 1138 . . . . . . . . 9  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  K { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } h )
84 elecg 6943 . . . . . . . . . 10  |-  ( ( h  e.  ( P pSyl 
G )  /\  K  e.  ( P pSyl  G ) )  ->  ( h  e.  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  <-> 
K { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } h ) )
8560, 59, 84syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  ( h  e.  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  <-> 
K { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } h ) )
8683, 85mpbird 224 . . . . . . . 8  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  h  e.  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )
8786ex 424 . . . . . . 7  |-  ( ph  ->  ( h  e.  ( P pSyl  G )  ->  h  e.  [ K ] { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } ) )
8887ssrdv 3354 . . . . . 6  |-  ( ph  ->  ( P pSyl  G ) 
C_  [ K ] { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )
8958, 88eqssd 3365 . . . . 5  |-  ( ph  ->  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  =  ( P pSyl  G
) )
9089fveq2d 5732 . . . 4  |-  ( ph  ->  ( # `  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )  =  ( # `  ( P pSyl  G ) ) )
915, 17, 1, 45, 19, 46, 47, 49, 50, 18sylow3lem2 15262 . . . . 5  |-  ( ph  ->  H  =  N )
9291fveq2d 5732 . . . 4  |-  ( ph  ->  ( # `  H
)  =  ( # `  N ) )
9390, 92oveq12d 6099 . . 3  |-  ( ph  ->  ( ( # `  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )  x.  ( # `  H
) )  =  ( ( # `  ( P pSyl  G ) )  x.  ( # `  N
) ) )
9454, 55, 933eqtr3rd 2477 . 2  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  x.  ( # `  N
) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) )
9516, 30, 43, 44, 94mulcan2ad 9658 1  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {cpr 3815   class class class wbr 4212   {copab 4265    e. cmpt 4266   ran crn 4879   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083    Er wer 6902   [cec 6903   /.cqs 6904   Fincfn 7109    x. cmul 8995   NNcn 10000   NN0cn0 10221   #chash 11618   Primecprime 13079   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   -gcsg 14688  SubGrpcsubg 14938   ~QG cqg 14940    GrpAct cga 15066   pSyl cslw 15166
This theorem is referenced by:  sylow3lem4  15264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-eqg 14943  df-ghm 15004  df-ga 15067  df-od 15167  df-pgp 15169  df-slw 15170
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