MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sylow3lem3 Unicode version

Theorem sylow3lem3 14940
Description: Lemma for sylow3 14944, 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 14938 . . . 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 2283 . . . . 5  |-  ( G ~QG  H )  =  ( G ~QG  H )
12 eqid 2283 . . . . 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 14768 . . . 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 2283 . . . 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 14658 . . . . 5  |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
)
182, 17syl 15 . . . 4  |-  ( ph  ->  N  e.  (SubGrp `  G ) )
191, 15, 18, 3lagsubg2 14678 . . 3  |-  ( ph  ->  ( # `  X
)  =  ( (
# `  ( X /. ( G ~QG  N ) ) )  x.  ( # `  N
) ) )
2012, 1gaorber 14762 . . . . . . . 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 6701 . . . . . 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 14937 . . . . . . . . . . 11  |-  ( (
ph  /\  h  e.  ( P pSyl  G )
)  ->  E. u  e.  X  h  =  ran  ( z  e.  K  |->  ( ( u  .+  z )  .-  u
) ) )
27 eqcom 2285 . . . . . . . . . . . . 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 5745 . . . . . . . . . . . . . . . . 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 4940 . . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  u  /\  y  =  K )  ->  ( x  .+  z
)  =  ( u 
.+  z ) )
3736, 35oveq12d 5876 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  =  u  /\  y  =  K )  ->  ( ( x  .+  z )  .-  x
)  =  ( ( u  .+  z ) 
.-  u ) )
3834, 37mpteq12dv 4098 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  u  /\  y  =  K )  ->  ( z  e.  y 
|->  ( ( x  .+  z )  .-  x
) )  =  ( z  e.  K  |->  ( ( u  .+  z
)  .-  u )
) )
3938rneqd 4906 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  u  /\  y  =  K )  ->  ran  ( z  e.  y  |->  ( ( x 
.+  z )  .-  x ) )  =  ran  ( z  e.  K  |->  ( ( u 
.+  z )  .-  u ) ) )
4039, 7ovmpt2ga 5977 . . . . . . . . . . . . . . 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 2294 . . . . . . . . . . . . 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 2560 . . . . . . . . . . 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 14761 . . . . . . . . . 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 6698 . . . . . . . . . 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 3185 . . . . . 6  |-  ( ph  ->  ( P pSyl  G ) 
C_  [ K ] { <. x ,  y
>.  |  ( {
x ,  y } 
C_  ( P pSyl  G
)  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) } )
5322, 52eqssd 3196 . . . . 5  |-  ( ph  ->  [ K ] { <. x ,  y >.  |  ( { x ,  y }  C_  ( P pSyl  G )  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }  =  ( P pSyl  G
) )
5453fveq2d 5529 . . . 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 14939 . . . . 5  |-  ( ph  ->  H  =  N )
5655fveq2d 5529 . . . 4  |-  ( ph  ->  ( # `  H
)  =  ( # `  N ) )
5754, 56oveq12d 5876 . . 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 2324 . 2  |-  ( ph  ->  ( ( # `  ( P pSyl  G ) )  x.  ( # `  N
) )  =  ( ( # `  ( X /. ( G ~QG  N ) ) )  x.  ( # `
 N ) ) )
59 pwfi 7151 . . . . . . 7  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
603, 59sylib 188 . . . . . 6  |-  ( ph  ->  ~P X  e.  Fin )
61 slwsubg 14921 . . . . . . . . 9  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  (SubGrp `  G ) )
621subgss 14622 . . . . . . . . 9  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
6361, 62syl 15 . . . . . . . 8  |-  ( x  e.  ( P pSyl  G
)  ->  x  C_  X
)
64 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
6564elpw 3631 . . . . . . . 8  |-  ( x  e.  ~P X  <->  x  C_  X
)
6663, 65sylibr 203 . . . . . . 7  |-  ( x  e.  ( P pSyl  G
)  ->  x  e.  ~P X )
6766ssriv 3184 . . . . . 6  |-  ( P pSyl 
G )  C_  ~P X
68 ssfi 7083 . . . . . 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 11350 . . . . 5  |-  ( ( P pSyl  G )  e. 
Fin  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
7169, 70syl 15 . . . 4  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e. 
NN0 )
7271nn0cnd 10020 . . 3  |-  ( ph  ->  ( # `  ( P pSyl  G ) )  e.  CC )
731, 15eqger 14667 . . . . . . . 8  |-  ( N  e.  (SubGrp `  G
)  ->  ( G ~QG  N
)  Er  X )
7418, 73syl 15 . . . . . . 7  |-  ( ph  ->  ( G ~QG  N )  Er  X
)
7574qsss 6720 . . . . . 6  |-  ( ph  ->  ( X /. ( G ~QG  N ) )  C_  ~P X )
76 ssfi 7083 . . . . . 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 11350 . . . . 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 10020 . . 3  |-  ( ph  ->  ( # `  ( X /. ( G ~QG  N ) ) )  e.  CC )
81 eqid 2283 . . . . . . . 8  |-  ( 0g
`  G )  =  ( 0g `  G
)
8281subg0cl 14629 . . . . . . 7  |-  ( N  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  N
)
8318, 82syl 15 . . . . . 6  |-  ( ph  ->  ( 0g `  G
)  e.  N )
84 ne0i 3461 . . . . . 6  |-  ( ( 0g `  G )  e.  N  ->  N  =/=  (/) )
8583, 84syl 15 . . . . 5  |-  ( ph  ->  N  =/=  (/) )
861subgss 14622 . . . . . . . 8  |-  ( N  e.  (SubGrp `  G
)  ->  N  C_  X
)
8718, 86syl 15 . . . . . . 7  |-  ( ph  ->  N  C_  X )
88 ssfi 7083 . . . . . . 7  |-  ( ( X  e.  Fin  /\  N  C_  X )  ->  N  e.  Fin )
893, 87, 88syl2anc 642 . . . . . 6  |-  ( ph  ->  N  e.  Fin )
90 hashnncl 11354 . . . . . 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 9762 . . 3  |-  ( ph  ->  ( # `  N
)  e.  CC )
9492nnne0d 9790 . . 3  |-  ( ph  ->  ( # `  N
)  =/=  0 )
9572, 80, 93, 94mulcan2d 9402 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {cpr 3641   class class class wbr 4023   {copab 4076    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    Er wer 6657   [cec 6658   /.cqs 6659   Fincfn 6863    x. cmul 8742   NNcn 9746   NN0cn0 9965   #chash 11337   Primecprime 12758   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615   ~QG cqg 14617    GrpAct cga 14743   pSyl cslw 14843
This theorem is referenced by:  sylow3lem4  14941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-ghm 14681  df-ga 14744  df-od 14844  df-pgp 14846  df-slw 14847
  Copyright terms: Public domain W3C validator