Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diophrex Structured version   Unicode version

Theorem diophrex 35588
Description: Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
diophrex  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
Distinct variable groups:    t, N, u    t, S, u
Allowed substitution hints:    M( u, t)

Proof of Theorem diophrex
Dummy variables  a 
b  c  d  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2426 . . . . 5  |-  ( a  =  t  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( b  |`  (
1 ... N ) ) ) )
21rexbidv 2936 . . . 4  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. b  e.  S  t  =  ( b  |`  (
1 ... N ) ) ) )
3 reseq1 5118 . . . . . 6  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
43eqeq2d 2436 . . . . 5  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
54cbvrexv 3055 . . . 4  |-  ( E. b  e.  S  t  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) )
62, 5syl6bb 264 . . 3  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) ) )
76cbvabv 2561 . 2  |-  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  =  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) }
8 rexeq 3023 . . . . . 6  |-  ( S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) }  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. b  e.  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } a  =  ( b  |`  ( 1 ... N
) ) ) )
98abbidv 2553 . . . . 5  |-  ( S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) }  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  =  { a  |  E. b  e. 
{ d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } a  =  ( b  |`  ( 1 ... N
) ) } )
109adantl 467 . . . 4  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M ) )  /\  c  e.  (mzPoly `  NN ) )  /\  S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } )  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) ) }  =  { a  |  E. b  e.  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) ) } )
11 eqeq1 2426 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
d  =  ( e  |`  ( 1 ... M
) )  <->  b  =  ( e  |`  (
1 ... M ) ) ) )
1211anbi1d 709 . . . . . . . . . 10  |-  ( d  =  b  ->  (
( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) ) )
1312rexbidv 2936 . . . . . . . . 9  |-  ( d  =  b  ->  ( E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  <->  E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) ) )
1413rexab 3233 . . . . . . . 8  |-  ( E. b  e.  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) )  <->  E. b ( E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) ) )
15 r19.41v 2977 . . . . . . . . . 10  |-  ( E. e  e.  ( NN0 
^m  NN ) ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) ) )
1615exbii 1712 . . . . . . . . 9  |-  ( E. b E. e  e.  ( NN0  ^m  NN ) ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  E. b
( E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) ) )
17 rexcom4 3101 . . . . . . . . . 10  |-  ( E. e  e.  ( NN0 
^m  NN ) E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  E. b E. e  e.  ( NN0  ^m  NN ) ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) ) )
18 anass 653 . . . . . . . . . . . . . 14  |-  ( ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) ) )
1918exbii 1712 . . . . . . . . . . . . 13  |-  ( E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  E. b
( b  =  ( e  |`  ( 1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) ) )
20 vex 3083 . . . . . . . . . . . . . . 15  |-  e  e. 
_V
2120resex 5167 . . . . . . . . . . . . . 14  |-  ( e  |`  ( 1 ... M
) )  e.  _V
22 reseq1 5118 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
b  |`  ( 1 ... N ) )  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
2322eqeq2d 2436 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
2423anbi2d 708 . . . . . . . . . . . . . 14  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
( ( c `  e )  =  0  /\  a  =  ( b  |`  ( 1 ... N ) ) )  <->  ( ( c `
 e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) ) )
2521, 24ceqsexv 3118 . . . . . . . . . . . . 13  |-  ( E. b ( b  =  ( e  |`  (
1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) )  <->  ( (
c `  e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
2619, 25bitri 252 . . . . . . . . . . . 12  |-  ( E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  ( (
c `  e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
27 ancom 451 . . . . . . . . . . . . 13  |-  ( ( ( c `  e
)  =  0  /\  a  =  ( ( e  |`  ( 1 ... M ) )  |`  ( 1 ... N
) ) )  <->  ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) )
28 simpl2 1009 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  M  e.  (
ZZ>= `  N ) )
29 fzss2 11846 . . . . . . . . . . . . . . . 16  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... M
) )
30 resabs1 5152 . . . . . . . . . . . . . . . 16  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( e  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( e  |`  ( 1 ... N ) ) )
3128, 29, 303syl 18 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  =  ( e  |`  ( 1 ... N
) ) )
3231eqeq2d 2436 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  <-> 
a  =  ( e  |`  ( 1 ... N
) ) ) )
3332anbi1d 709 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 )  <->  ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3427, 33syl5bb 260 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( ( ( c `  e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )  <->  ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3526, 34syl5bb 260 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. b
( ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3635rexbidv 2936 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. e  e.  ( NN0  ^m  NN ) E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3717, 36syl5bbr 262 . . . . . . . . 9  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. b E. e  e.  ( NN0  ^m  NN ) ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  ( 1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3816, 37syl5bbr 262 . . . . . . . 8  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. b
( E. e  e.  ( NN0  ^m  NN ) ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  ( 1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
3914, 38syl5bb 260 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( E. b  e.  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } a  =  ( b  |`  ( 1 ... N
) )  <->  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) ) )
4039abbidv 2553 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. b  e.  {
d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) ) }  =  { a  |  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) } )
41 eldioph3 35578 . . . . . . 7  |-  ( ( N  e.  NN0  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) }  e.  (Dioph `  N ) )
42413ad2antl1 1167 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  ( 1 ... N ) )  /\  ( c `  e )  =  0 ) }  e.  (Dioph `  N ) )
4340, 42eqeltrd 2507 . . . . 5  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. b  e.  {
d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) ) }  e.  (Dioph `  N ) )
4443adantr 466 . . . 4  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M ) )  /\  c  e.  (mzPoly `  NN ) )  /\  S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } )  ->  { a  |  E. b  e.  {
d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) } a  =  ( b  |`  (
1 ... N ) ) }  e.  (Dioph `  N ) )
4510, 44eqeltrd 2507 . . 3  |-  ( ( ( ( N  e. 
NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M ) )  /\  c  e.  (mzPoly `  NN ) )  /\  S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 ) } )  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
46 eldioph3b 35577 . . . . 5  |-  ( S  e.  (Dioph `  M
)  <->  ( M  e. 
NN0  /\  E. c  e.  (mzPoly `  NN ) S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) } ) )
4746simprbi 465 . . . 4  |-  ( S  e.  (Dioph `  M
)  ->  E. c  e.  (mzPoly `  NN ) S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) } )
48473ad2ant3 1028 . . 3  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  E. c  e.  (mzPoly `  NN ) S  =  { d  |  E. e  e.  ( NN0  ^m  NN ) ( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 ) } )
4945, 48r19.29a 2967 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
507, 49syl5eqelr 2512 1  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407   E.wrex 2772    C_ wss 3436    |` cres 4855   ` cfv 5601  (class class class)co 6306    ^m cmap 7484   0cc0 9547   1c1 9548   NNcn 10617   NN0cn0 10877   ZZ>=cuz 11167   ...cfz 11792  mzPolycmzp 35534  Diophcdioph 35567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-of 6546  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-oadd 7198  df-er 7375  df-map 7486  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-card 8382  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-n0 10878  df-z 10946  df-uz 11168  df-fz 11793  df-hash 12523  df-mzpcl 35535  df-mzp 35536  df-dioph 35568
This theorem is referenced by:  rexrabdioph  35607
  Copyright terms: Public domain W3C validator