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

Theorem dprdss 15264
Description: Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdss.1  |-  ( ph  ->  G dom DProd  T )
dprdss.2  |-  ( ph  ->  dom  T  =  I )
dprdss.3  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
dprdss.4  |-  ( (
ph  /\  k  e.  I )  ->  ( S `  k )  C_  ( T `  k
) )
Assertion
Ref Expression
dprdss  |-  ( ph  ->  ( G dom DProd  S  /\  ( G DProd  S )  C_  ( G DProd  T ) ) )
Distinct variable groups:    k, G    ph, k    S, k    T, k   
k, I

Proof of Theorem dprdss
Dummy variables  f 
a  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2283 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2283 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdss.1 . . . 4  |-  ( ph  ->  G dom DProd  T )
5 dprdgrp 15240 . . . 4  |-  ( G dom DProd  T  ->  G  e. 
Grp )
64, 5syl 15 . . 3  |-  ( ph  ->  G  e.  Grp )
7 dprdss.2 . . . 4  |-  ( ph  ->  dom  T  =  I )
8 reldmdprd 15235 . . . . . 6  |-  Rel  dom DProd
98brrelex2i 4730 . . . . 5  |-  ( G dom DProd  T  ->  T  e. 
_V )
10 dmexg 4939 . . . . 5  |-  ( T  e.  _V  ->  dom  T  e.  _V )
114, 9, 103syl 18 . . . 4  |-  ( ph  ->  dom  T  e.  _V )
127, 11eqeltrrd 2358 . . 3  |-  ( ph  ->  I  e.  _V )
13 dprdss.3 . . 3  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
14 dprdss.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  I )  ->  ( S `  k )  C_  ( T `  k
) )
1514ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
16 fveq2 5525 . . . . . . . 8  |-  ( k  =  x  ->  ( S `  k )  =  ( S `  x ) )
17 fveq2 5525 . . . . . . . 8  |-  ( k  =  x  ->  ( T `  k )  =  ( T `  x ) )
1816, 17sseq12d 3207 . . . . . . 7  |-  ( k  =  x  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  x )  C_  ( T `  x )
) )
1918rspcv 2880 . . . . . 6  |-  ( x  e.  I  ->  ( A. k  e.  I 
( S `  k
)  C_  ( T `  k )  ->  ( S `  x )  C_  ( T `  x
) ) )
2015, 19mpan9 455 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  ( T `  x
) )
21203ad2antr1 1120 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( T `  x ) )
224adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  G dom DProd  T )
237adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  dom  T  =  I )
24 simpr1 961 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  x  e.  I )
25 simpr2 962 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
y  e.  I )
26 simpr3 963 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  x  =/=  y )
2722, 23, 24, 25, 26, 1dprdcntz 15243 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( T `  y )
) )
284, 7dprdf2 15242 . . . . . . . . 9  |-  ( ph  ->  T : I --> (SubGrp `  G ) )
2928adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  T : I --> (SubGrp `  G ) )
30 ffvelrn 5663 . . . . . . . 8  |-  ( ( T : I --> (SubGrp `  G )  /\  y  e.  I )  ->  ( T `  y )  e.  (SubGrp `  G )
)
3129, 25, 30syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  e.  (SubGrp `  G ) )
32 eqid 2283 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
3332subgss 14622 . . . . . . 7  |-  ( ( T `  y )  e.  (SubGrp `  G
)  ->  ( T `  y )  C_  ( Base `  G ) )
3431, 33syl 15 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  C_  ( Base `  G ) )
3515adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
36 fveq2 5525 . . . . . . . . 9  |-  ( k  =  y  ->  ( S `  k )  =  ( S `  y ) )
37 fveq2 5525 . . . . . . . . 9  |-  ( k  =  y  ->  ( T `  k )  =  ( T `  y ) )
3836, 37sseq12d 3207 . . . . . . . 8  |-  ( k  =  y  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  y )  C_  ( T `  y )
) )
3938rspcv 2880 . . . . . . 7  |-  ( y  e.  I  ->  ( A. k  e.  I 
( S `  k
)  C_  ( T `  k )  ->  ( S `  y )  C_  ( T `  y
) ) )
4025, 35, 39sylc 56 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  y
)  C_  ( T `  y ) )
4132, 1cntz2ss 14808 . . . . . 6  |-  ( ( ( T `  y
)  C_  ( Base `  G )  /\  ( S `  y )  C_  ( T `  y
) )  ->  (
(Cntz `  G ) `  ( T `  y
) )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
4234, 40, 41syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( T `  y ) )  C_  ( (Cntz `  G ) `  ( S `  y
) ) )
4327, 42sstrd 3189 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
4421, 43sstrd 3189 . . 3  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
456adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
4632subgacs 14652 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
47 acsmre 13554 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
4845, 46, 473syl 18 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
49 difss 3303 . . . . . . . . 9  |-  ( I 
\  { x }
)  C_  I
5015adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  I  ( S `  k )  C_  ( T `  k )
)
51 ssralv 3237 . . . . . . . . 9  |-  ( ( I  \  { x } )  C_  I  ->  ( A. k  e.  I  ( S `  k )  C_  ( T `  k )  ->  A. k  e.  ( I  \  { x } ) ( S `
 k )  C_  ( T `  k ) ) )
5249, 50, 51mpsyl 59 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  ( T `  k ) )
53 ss2iun 3920 . . . . . . . 8  |-  ( A. k  e.  ( I  \  { x } ) ( S `  k
)  C_  ( T `  k )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  U_ k  e.  ( I  \  { x } ) ( T `
 k ) )
5452, 53syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  U_ k  e.  ( I  \  { x } ) ( T `
 k ) )
5513adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
56 ffun 5391 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  Fun  S )
57 funiunfv 5774 . . . . . . . 8  |-  ( Fun 
S  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k )  =  U. ( S
" ( I  \  { x } ) ) )
5855, 56, 573syl 18 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k )  =  U. ( S
" ( I  \  { x } ) ) )
5928adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  T : I --> (SubGrp `  G ) )
60 ffun 5391 . . . . . . . 8  |-  ( T : I --> (SubGrp `  G )  ->  Fun  T )
61 funiunfv 5774 . . . . . . . 8  |-  ( Fun 
T  ->  U_ k  e.  ( I  \  {
x } ) ( T `  k )  =  U. ( T
" ( I  \  { x } ) ) )
6259, 60, 613syl 18 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( T `  k )  =  U. ( T
" ( I  \  { x } ) ) )
6354, 58, 623sstr3d 3220 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  U. ( T " ( I  \  { x } ) ) )
64 imassrn 5025 . . . . . . . 8  |-  ( T
" ( I  \  { x } ) )  C_  ran  T
65 frn 5395 . . . . . . . . . 10  |-  ( T : I --> (SubGrp `  G )  ->  ran  T 
C_  (SubGrp `  G )
)
6659, 65syl 15 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  (SubGrp `  G )
)
67 mresspw 13494 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6848, 67syl 15 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6966, 68sstrd 3189 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  ~P ( Base `  G
) )
7064, 69syl5ss 3190 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( T " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
71 sspwuni 3987 . . . . . . 7  |-  ( ( T " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( T " (
I  \  { x } ) )  C_  ( Base `  G )
)
7270, 71sylib 188 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( T " ( I  \  { x } ) )  C_  ( Base `  G ) )
733mrcss 13518 . . . . . 6  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " (
I  \  { x } ) )  C_  U. ( T " (
I  \  { x } ) )  /\  U. ( T " (
I  \  { x } ) )  C_  ( Base `  G )
)  ->  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) )  C_  (
(mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )
7448, 63, 72, 73syl3anc 1182 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )
75 ss2in 3396 . . . . 5  |-  ( ( ( S `  x
)  C_  ( T `  x )  /\  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )  ->  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  C_  ( ( T `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( T " (
I  \  { x } ) ) ) ) )
7620, 74, 75syl2anc 642 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  ( ( T `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) ) )
774adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  G dom DProd  T )
787adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  dom  T  =  I )
79 simpr 447 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
8077, 78, 79, 2, 3dprddisj 15244 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( T `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( T
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
8176, 80sseqtrd 3214 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
821, 2, 3, 6, 12, 13, 44, 81dmdprdd 15237 . 2  |-  ( ph  ->  G dom DProd  S )
834a1d 22 . . . . 5  |-  ( ph  ->  ( G dom DProd  S  ->  G dom DProd  T ) )
84 ss2ixp 6829 . . . . . . 7  |-  ( A. k  e.  I  ( S `  k )  C_  ( T `  k
)  ->  X_ k  e.  I  ( S `  k )  C_  X_ k  e.  I  ( T `  k ) )
8515, 84syl 15 . . . . . 6  |-  ( ph  -> 
X_ k  e.  I 
( S `  k
)  C_  X_ k  e.  I  ( T `  k ) )
86 rabss2 3256 . . . . . 6  |-  ( X_ k  e.  I  ( S `  k )  C_  X_ k  e.  I 
( T `  k
)  ->  { h  e.  X_ k  e.  I 
( S `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
C_  { h  e.  X_ k  e.  I 
( T `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )
87 ssrexv 3238 . . . . . 6  |-  ( { h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
C_  { h  e.  X_ k  e.  I 
( T `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }  ->  ( E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f )  ->  E. f  e.  {
h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } a  =  ( G 
gsumg  f ) ) )
8885, 86, 873syl 18 . . . . 5  |-  ( ph  ->  ( E. f  e. 
{ h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f )  ->  E. f  e.  {
h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } a  =  ( G 
gsumg  f ) ) )
8983, 88anim12d 546 . . . 4  |-  ( ph  ->  ( ( G dom DProd  S  /\  E. f  e. 
{ h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) )  ->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
90 fdm 5393 . . . . 5  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
91 eqid 2283 . . . . . 6  |-  { h  e.  X_ k  e.  I 
( S `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }  =  { h  e.  X_ k  e.  I 
( S `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }
922, 91eldprd 15239 . . . . 5  |-  ( dom 
S  =  I  -> 
( a  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
9313, 90, 923syl 18 . . . 4  |-  ( ph  ->  ( a  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
94 eqid 2283 . . . . . 6  |-  { h  e.  X_ k  e.  I 
( T `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }  =  { h  e.  X_ k  e.  I 
( T `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }
952, 94eldprd 15239 . . . . 5  |-  ( dom 
T  =  I  -> 
( a  e.  ( G DProd  T )  <->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
967, 95syl 15 . . . 4  |-  ( ph  ->  ( a  e.  ( G DProd  T )  <->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
9789, 93, 963imtr4d 259 . . 3  |-  ( ph  ->  ( a  e.  ( G DProd  S )  -> 
a  e.  ( G DProd 
T ) ) )
9897ssrdv 3185 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  ( G DProd  T
) )
9982, 98jca 518 1  |-  ( ph  ->  ( G dom DProd  S  /\  ( G DProd  S )  C_  ( G DProd  T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   U_ciun 3905   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   X_cixp 6817   Fincfn 6863   Basecbs 13148   0gc0g 13400    gsumg cgsu 13401  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
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-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
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-iin 3908  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-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-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-oadd 6483  df-er 6660  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-cntz 14793  df-dprd 15233
  Copyright terms: Public domain W3C validator