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Theorem diophrex 30871
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 2461 . . . . 5  |-  ( a  =  t  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( b  |`  (
1 ... N ) ) ) )
21rexbidv 2968 . . . 4  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. b  e.  S  t  =  ( b  |`  (
1 ... N ) ) ) )
3 reseq1 5277 . . . . . 6  |-  ( b  =  u  ->  (
b  |`  ( 1 ... N ) )  =  ( u  |`  (
1 ... N ) ) )
43eqeq2d 2471 . . . . 5  |-  ( b  =  u  ->  (
t  =  ( b  |`  ( 1 ... N
) )  <->  t  =  ( u  |`  ( 1 ... N ) ) ) )
54cbvrexv 3085 . . . 4  |-  ( E. b  e.  S  t  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) )
62, 5syl6bb 261 . . 3  |-  ( a  =  t  ->  ( E. b  e.  S  a  =  ( b  |`  ( 1 ... N
) )  <->  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) ) )
76cbvabv 2600 . 2  |-  { a  |  E. b  e.  S  a  =  ( b  |`  ( 1 ... N ) ) }  =  { t  |  E. u  e.  S  t  =  ( u  |`  ( 1 ... N ) ) }
8 rexeq 3055 . . . . . 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 2593 . . . . 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 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.  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 2461 . . . . . . . . . . 11  |-  ( d  =  b  ->  (
d  =  ( e  |`  ( 1 ... M
) )  <->  b  =  ( e  |`  (
1 ... M ) ) ) )
1211anbi1d 704 . . . . . . . . . 10  |-  ( d  =  b  ->  (
( d  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( c `  e )  =  0 ) ) )
1312rexbidv 2968 . . . . . . . . 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 3262 . . . . . . . 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 3009 . . . . . . . . . 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 1668 . . . . . . . . 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 3129 . . . . . . . . . 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 649 . . . . . . . . . . . . . 14  |-  ( ( ( b  =  ( e  |`  ( 1 ... M ) )  /\  ( c `  e )  =  0 )  /\  a  =  ( b  |`  (
1 ... N ) ) )  <->  ( b  =  ( e  |`  (
1 ... M ) )  /\  ( ( c `
 e )  =  0  /\  a  =  ( b  |`  (
1 ... N ) ) ) ) )
1918exbii 1668 . . . . . . . . . . . . 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 3112 . . . . . . . . . . . . . . 15  |-  e  e. 
_V
2120resex 5327 . . . . . . . . . . . . . 14  |-  ( e  |`  ( 1 ... M
) )  e.  _V
22 reseq1 5277 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
b  |`  ( 1 ... N ) )  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
2322eqeq2d 2471 . . . . . . . . . . . . . . 15  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
a  =  ( b  |`  ( 1 ... N
) )  <->  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) )
2423anbi2d 703 . . . . . . . . . . . . . 14  |-  ( b  =  ( e  |`  ( 1 ... M
) )  ->  (
( ( c `  e )  =  0  /\  a  =  ( b  |`  ( 1 ... N ) ) )  <->  ( ( c `
 e )  =  0  /\  a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) ) ) )
2521, 24ceqsexv 3146 . . . . . . . . . . . . 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 249 . . . . . . . . . . . 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 450 . . . . . . . . . . . . 13  |-  ( ( ( c `  e
)  =  0  /\  a  =  ( ( e  |`  ( 1 ... M ) )  |`  ( 1 ... N
) ) )  <->  ( a  =  ( ( e  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /\  ( c `  e )  =  0 ) )
28 simpl2 1000 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  M  e.  ( ZZ>= `  N )  /\  S  e.  (Dioph `  M )
)  /\  c  e.  (mzPoly `  NN ) )  ->  M  e.  (
ZZ>= `  N ) )
29 fzss2 11748 . . . . . . . . . . . . . . . 16  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 1 ... N )  C_  ( 1 ... M
) )
30 resabs1 5312 . . . . . . . . . . . . . . . 16  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( e  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( e  |`  ( 1 ... N ) ) )
3128, 29, 303syl 20 . . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . . 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 704 . . . . . . . . . . . . 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 257 . . . . . . . . . . . 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 257 . . . . . . . . . . 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 2968 . . . . . . . . . 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 259 . . . . . . . . 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 259 . . . . . . . 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 257 . . . . . . 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 2593 . . . . . 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 30861 . . . . . . 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 1158 . . . . . 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 2545 . . . . 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 465 . . . 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 2545 . . 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 30860 . . . . 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 464 . . . 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 1019 . . 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 2999 . 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 2550 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 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442   E.wrex 2808    C_ wss 3471    |` cres 5010   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   0cc0 9509   1c1 9510   NNcn 10556   NN0cn0 10816   ZZ>=cuz 11106   ...cfz 11697  mzPolycmzp 30816  Diophcdioph 30850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12408  df-mzpcl 30817  df-mzp 30818  df-dioph 30851
This theorem is referenced by:  rexrabdioph  30889
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