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Theorem diophrex 26855
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 2289 . . . . 5  |-  ( a  =  t  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( b  |`  (
1 ... N ) ) ) )
21rexbidv 2564 . . . 4  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. b  e.  S  t  =  ( b  |`  (
1 ... N ) ) ) )
3 reseq1 4949 . . . . . 6  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
43eqeq2d 2294 . . . . 5  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
54cbvrexv 2765 . . . 4  |-  ( E. b  e.  S  t  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) )
62, 5syl6bb 252 . . 3  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) ) )
76cbvabv 2402 . 2  |-  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  =  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) }
8 eldioph3b 26844 . . . . 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 ) } ) )
98simprbi 450 . . . 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 ) } )
1093ad2ant3 978 . . 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 ) } )
11 rexeq 2737 . . . . . . . 8  |-  ( 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
) ) ) )
1211abbidv 2397 . . . . . . 7  |-  ( 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
) ) } )
1312adantl 452 . . . . . 6  |-  ( ( ( ( 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 ) ) } )
14 eqeq1 2289 . . . . . . . . . . . . 13  |-  ( d  =  b  ->  (
d  =  ( e  |`  ( 1 ... M
) )  <->  b  =  ( e  |`  (
1 ... M ) ) ) )
1514anbi1d 685 . . . . . . . . . . . 12  |-  ( d  =  b  ->  (
( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) ) )
1615rexbidv 2564 . . . . . . . . . . 11  |-  ( 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 ) ) )
1716rexab 2928 . . . . . . . . . 10  |-  ( 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 ) ) ) )
18 r19.41v 2693 . . . . . . . . . . . 12  |-  ( 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 ) ) ) )
1918exbii 1569 . . . . . . . . . . 11  |-  ( 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 ) ) ) )
20 rexcom4 2807 . . . . . . . . . . . 12  |-  ( 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 ) ) ) )
21 anass 630 . . . . . . . . . . . . . . . 16  |-  ( ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) ) )
2221exbii 1569 . . . . . . . . . . . . . . 15  |-  ( 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 ) ) ) ) )
23 vex 2791 . . . . . . . . . . . . . . . . 17  |-  e  e. 
_V
2423resex 4995 . . . . . . . . . . . . . . . 16  |-  ( e  |`  ( 1 ... M
) )  e.  _V
25 reseq1 4949 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
b  |`  ( 1 ... N ) )  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
2625eqeq2d 2294 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
2726anbi2d 684 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
( ( c `  e )  =  0  /\  a  =  ( b  |`  ( 1 ... N ) ) )  <->  ( ( c `
 e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) ) )
2824, 27ceqsexv 2823 . . . . . . . . . . . . . . 15  |-  ( E. b ( b  =  ( e  |`  (
1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) )  <->  ( (
c `  e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
2922, 28bitri 240 . . . . . . . . . . . . . 14  |-  ( E. b ( ( b  =  ( e  |`  ( 1 ... M
) )  /\  (
c `  e )  =  0 )  /\  a  =  ( b  |`  ( 1 ... N
) ) )  <->  ( (
c `  e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
30 ancom 437 . . . . . . . . . . . . . . 15  |-  ( ( ( c `  e
)  =  0  /\  a  =  ( ( e  |`  ( 1 ... M ) )  |`  ( 1 ... N
) ) )  <->  ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) )
31 simpl2 959 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  M  e.  (
ZZ>= `  N ) )
32 fzss2 10831 . . . . . . . . . . . . . . . . . 18  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... M
) )
33 resabs1 4984 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( e  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( e  |`  ( 1 ... N ) ) )
3431, 32, 333syl 18 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  =  ( e  |`  ( 1 ... N
) ) )
3534eqeq2d 2294 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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
) ) ) )
3635anbi1d 685 . . . . . . . . . . . . . . 15  |-  ( ( ( 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 ) ) )
3730, 36syl5bb 248 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )
3829, 37syl5bb 248 . . . . . . . . . . . . 13  |-  ( ( ( 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 ) ) )
3938rexbidv 2564 . . . . . . . . . . . 12  |-  ( ( ( 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 ) ) )
4020, 39syl5bbr 250 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) )
4119, 40syl5bbr 250 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
4217, 41syl5bb 248 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
4342abbidv 2397 . . . . . . . 8  |-  ( ( ( 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 ) } )
44 eldioph3 26845 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  c  e.  (mzPoly `  NN ) )  ->  { a  |  E. e  e.  ( NN0  ^m  NN ) ( a  =  ( e  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) }  e.  (Dioph `  N ) )
45443ad2antl1 1117 . . . . . . . 8  |-  ( ( ( 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 ) )
4643, 45eqeltrd 2357 . . . . . . 7  |-  ( ( ( 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 ) )
4746adantr 451 . . . . . 6  |-  ( ( ( ( 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 ) )
4813, 47eqeltrd 2357 . . . . 5  |-  ( ( ( ( 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 ) )
4948ex 423 . . . 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
) ) }  e.  (Dioph `  N ) ) )
5049rexlimdva 2667 . . 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 ) }  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  e.  (Dioph `  N ) ) )
5110, 50mpd 14 . 2  |-  ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  ->  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) ) }  e.  (Dioph `  N ) )
527, 51syl5eqelr 2368 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    C_ wss 3152    |` cres 4691   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   0cc0 8737   1c1 8738   NNcn 9746   NN0cn0 9965   ZZ>=cuz 10230   ...cfz 10782  mzPolycmzp 26800  Diophcdioph 26834
This theorem is referenced by:  rexrabdioph  26875
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
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-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-of 6078  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-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  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-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-hash 11338  df-mzpcl 26801  df-mzp 26802  df-dioph 26835
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