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Theorem sylow3lem3 14956
Description: Lemma for sylow3 14960, 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.x . . . . 5  |-  X  =  ( Base `  G
)
2 sylow3.g . . . . 5  |-  ( ph  ->  G  e.  Grp )
3 sylow3.xf . . . . 5  |-  ( ph  ->  X  e.  Fin )
4 sylow3.p . . . . 5  |-  ( ph  ->  P  e.  Prime )
5 sylow3lem1.a . . . . 5  |-  .+  =  ( +g  `  G )
6 sylow3lem1.d . . . . 5  |-  .-  =  ( -g `  G )
7 sylow3lem1.m . . . . 5  |-  .(+)  =  ( x  e.  X , 
y  e.  ( P pSyl 
G )  |->  ran  (
z  e.  y  |->  ( ( x  .+  z
)  .-  x )
) )
81, 2, 3, 4, 5, 6, 7sylow3lem1 14954 . . . 4  |-  ( ph  -> 
.(+)  e.  ( G  GrpAct  ( P pSyl  G )
) )
9 sylow3lem2.k . . . 4  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
10 sylow3lem2.h . . . . 5  |-  H  =  { u  e.  X  |  ( u  .(+)  K )  =  K }
11 eqid 2296 . . . . 5  |-  ( G ~QG  H )  =  ( G ~QG  H )
12 eqid 2296 . . . . 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 ) }
131, 10, 11, 12orbsta2 14784 . . . 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
) ) )
148, 9, 3, 13syl21anc 1181 . . 3  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  [ K ] { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )  x.  ( # `  H
) ) )
15 eqid 2296 . . . 4  |-  ( G ~QG  N )  =  ( G ~QG  N )
16 sylow3lem2.n . . . . . 6  |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x  .+  y )  e.  K  <->  ( y  .+  x )  e.  K ) }
1716, 1, 5nmzsubg 14674 . . . . 5  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
182, 17syl 15 . . . 4  |-  ( ph  ->  N  e.  (SubGrp `  G ) )
191, 15, 18, 3lagsubg2 14694 . . 3  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
2012, 1gaorber 14778 . . . . . . . 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 ) )
218, 20syl 15 . . . . . . 7  |-  ( ph  ->  { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  Er  ( P pSyl  G ) )
2221ecss 6717 . . . . . 6  |-  ( ph  ->  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } 
C_  ( P pSyl  G
) )
239adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  K  e.  ( P pSyl  G )
)
24 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  h  e.  ( P pSyl  G )
)
253adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  X  e.  Fin )
261, 25, 24, 23, 5, 6sylow2 14953 . . . . . . . . . . 11  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  E. u  e.  X  h  =  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) ) )
27 eqcom 2298 . . . . . . . . . . . . 13  |-  ( ( u  .(+)  K )  =  h  <->  h  =  (
u  .(+)  K ) )
28 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  u  e.  X )
2923adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  K  e.  ( P pSyl  G ) )
30 mptexg 5761 . . . . . . . . . . . . . . . . 17  |-  ( K  e.  ( P pSyl  G
)  ->  ( z  e.  K  |->  ( ( u  .+  z ) 
.-  u ) )  e.  _V )
3129, 30syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  (
z  e.  K  |->  ( ( u  .+  z
)  .-  u )
)  e.  _V )
32 rnexg 4956 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  K  |->  ( ( u  .+  z
)  .-  u )
)  e.  _V  ->  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) )  e.  _V )
3331, 32syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) )  e.  _V )
34 simpr 447 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  u  /\  y  =  K )  ->  y  =  K )
35 simpl 443 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  u  /\  y  =  K )  ->  x  =  u )
3635oveq1d 5889 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  u  /\  y  =  K )  ->  ( x  .+  z
)  =  ( u 
.+  z ) )
3736, 35oveq12d 5892 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  u  /\  y  =  K )  ->  ( ( x  .+  z )  .-  x
)  =  ( ( u  .+  z ) 
.-  u ) )
3834, 37mpteq12dv 4114 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  u  /\  y  =  K )  ->  ( z  e.  y 
|->  ( ( x  .+  z )  .-  x
) )  =  ( z  e.  K  |->  ( ( u  .+  z
)  .-  u )
) )
3938rneqd 4922 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  u  /\  y  =  K )  ->  ran  ( z  e.  y  |->  ( ( x 
.+  z )  .-  x ) )  =  ran  ( z  e.  K  |->  ( ( u 
.+  z )  .-  u ) ) )
4039, 7ovmpt2ga 5993 . . . . . . . . . . . . . . 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 ) ) )
4128, 29, 33, 40syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  (
u  .(+)  K )  =  ran  ( z  e.  K  |->  ( ( u 
.+  z )  .-  u ) ) )
4241eqeq2d 2307 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  (
h  =  ( u 
.(+)  K )  <->  h  =  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) ) ) )
4327, 42syl5bb 248 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  h  e.  ( P pSyl  G ) )  /\  u  e.  X )  ->  (
( u  .(+)  K )  =  h  <->  h  =  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) ) ) )
4443rexbidva 2573 . . . . . . . . . . 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 ) ) ) )
4526, 44mpbird 223 . . . . . . . . . 10  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  E. u  e.  X  ( u  .(+) 
K )  =  h )
4612gaorb 14777 . . . . . . . . . 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 ) )
4723, 24, 45, 46syl3anbrc 1136 . . . . . . . . 9  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  K { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } h )
48 elecg 6714 . . . . . . . . . 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 ) )
4924, 23, 48syl2anc 642 . . . . . . . . 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 ) )
5047, 49mpbird 223 . . . . . . . 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 ) } )
5150ex 423 . . . . . . 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 ) } ) )
5251ssrdv 3198 . . . . . 6  |-  ( ph  ->  ( P pSyl  G ) 
C_  [ K ] { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )
5322, 52eqssd 3209 . . . . 5  |-  ( ph  ->  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  =  ( P pSyl  G
) )
5453fveq2d 5545 . . . 4  |-  ( ph  ->  ( # `  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )  =  ( # `  ( P pSyl  G ) ) )
551, 2, 3, 4, 5, 6, 7, 9, 10, 16sylow3lem2 14955 . . . . 5  |-  ( ph  ->  H  =  N )
5655fveq2d 5545 . . . 4  |-  ( ph  ->  ( # `  H
)  =  ( # `  N ) )
5754, 56oveq12d 5892 . . 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
) ) )
5814, 19, 573eqtr3rd 2337 . 2  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  x.  ( # `  N
) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) )
59 pwfi 7167 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
603, 59sylib 188 . . . . . 6  |-  ( ph  ->  ~P X  e.  Fin )
61 slwsubg 14937 . . . . . . . . 9  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  (SubGrp `  G ) )
621subgss 14638 . . . . . . . . 9  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
6361, 62syl 15 . . . . . . . 8  |-  ( x  e.  ( P pSyl  G
)  ->  x  C_  X
)
64 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
6564elpw 3644 . . . . . . . 8  |-  ( x  e.  ~P X  <->  x  C_  X
)
6663, 65sylibr 203 . . . . . . 7  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  ~P X )
6766ssriv 3197 . . . . . 6  |-  ( P pSyl 
G )  C_  ~P X
68 ssfi 7099 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( P pSyl  G ) 
C_  ~P X )  -> 
( P pSyl  G )  e.  Fin )
6960, 67, 68sylancl 643 . . . . 5  |-  ( ph  ->  ( P pSyl  G )  e.  Fin )
70 hashcl 11366 . . . . 5  |-  ( ( P pSyl  G )  e. 
Fin  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
7169, 70syl 15 . . . 4  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
7271nn0cnd 10036 . . 3  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e.  CC )
731, 15eqger 14683 . . . . . . . 8  |-  ( N  e.  (SubGrp `  G
)  ->  ( G ~QG  N
)  Er  X )
7418, 73syl 15 . . . . . . 7  |-  ( ph  ->  ( G ~QG  N )  Er  X
)
7574qsss 6736 . . . . . 6  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  C_  ~P X )
76 ssfi 7099 . . . . . 6  |-  ( ( ~P X  e.  Fin  /\  ( X /. ( G ~QG  N ) )  C_  ~P X )  ->  ( X /. ( G ~QG  N ) )  e.  Fin )
7760, 75, 76syl2anc 642 . . . . 5  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  e. 
Fin )
78 hashcl 11366 . . . . 5  |-  ( ( X /. ( G ~QG  N ) )  e.  Fin  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
7977, 78syl 15 . . . 4  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  NN0 )
8079nn0cnd 10036 . . 3  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  CC )
81 eqid 2296 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
8281subg0cl 14645 . . . . . . 7  |-  ( N  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  N
)
8318, 82syl 15 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  e.  N )
84 ne0i 3474 . . . . . 6  |-  ( ( 0g `  G )  e.  N  ->  N  =/=  (/) )
8583, 84syl 15 . . . . 5  |-  ( ph  ->  N  =/=  (/) )
861subgss 14638 . . . . . . . 8  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
8718, 86syl 15 . . . . . . 7  |-  ( ph  ->  N  C_  X )
88 ssfi 7099 . . . . . . 7  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
893, 87, 88syl2anc 642 . . . . . 6  |-  ( ph  ->  N  e.  Fin )
90 hashnncl 11370 . . . . . 6  |-  ( N  e.  Fin  ->  (
( # `  N )  e.  NN  <->  N  =/=  (/) ) )
9189, 90syl 15 . . . . 5  |-  ( ph  ->  ( ( # `  N
)  e.  NN  <->  N  =/=  (/) ) )
9285, 91mpbird 223 . . . 4  |-  ( ph  ->  ( # `  N
)  e.  NN )
9392nncnd 9778 . . 3  |-  ( ph  ->  ( # `  N
)  e.  CC )
9492nnne0d 9806 . . 3  |-  ( ph  ->  ( # `  N
)  =/=  0 )
9572, 80, 93, 94mulcan2d 9418 . 2  |-  ( ph  ->  ( ( ( # `  ( P pSyl  G ) )  x.  ( # `  N ) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) )  <-> 
( # `  ( P pSyl 
G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) ) )
9658, 95mpbid 201 1  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  =  ( # `  ( X /. ( G ~QG  N ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {cpr 3654   class class class wbr 4039   {copab 4092    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    Er wer 6673   [cec 6674   /.cqs 6675   Fincfn 6879    x. cmul 8758   NNcn 9762   NN0cn0 9981   #chash 11353   Primecprime 12774   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   -gcsg 14381  SubGrpcsubg 14631   ~QG cqg 14633    GrpAct cga 14759   pSyl cslw 14859
This theorem is referenced by:  sylow3lem4  14957
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-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-1st 6138  df-2nd 6139  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-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-mnd 14383  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-ga 14760  df-od 14860  df-pgp 14862  df-slw 14863
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