Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  3rexfrabdioph Structured version   Unicode version

Theorem 3rexfrabdioph 30561
Description: Diophantine set builder for existential quantifier, explicit substitution, two variables. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
rexfrabdioph.1  |-  M  =  ( N  +  1 )
rexfrabdioph.2  |-  L  =  ( M  +  1 )
rexfrabdioph.3  |-  K  =  ( L  +  1 )
Assertion
Ref Expression
3rexfrabdioph  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  N ) )
Distinct variable groups:    u, t,
v, w, x, K   
t, L, u, v, w, x    t, M, u, v, w, x   
t, N, u, v, w, x    ph, t
Allowed substitution hints:    ph( x, w, v, u)

Proof of Theorem 3rexfrabdioph
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 sbc2rex 30551 . . . . . . 7  |-  ( [. ( a `  M
)  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  E. w  e.  NN0  E. x  e. 
NN0  [. ( a `  M )  /  v ]. ph )
21sbcbii 3391 . . . . . 6  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. E. w  e.  NN0  E. x  e.  NN0  [. (
a `  M )  /  v ]. ph )
3 sbc2rex 30551 . . . . . 6  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. E. w  e.  NN0  E. x  e.  NN0  [. ( a `  M )  /  v ]. ph  <->  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph )
42, 3bitri 249 . . . . 5  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph )
54a1i 11 . . . 4  |-  ( a  e.  ( NN0  ^m  ( 1 ... M
) )  ->  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph  <->  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph ) )
65rabbiia 3102 . . 3  |-  { a  e.  ( NN0  ^m  ( 1 ... M
) )  |  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph }  =  {
a  e.  ( NN0 
^m  ( 1 ... M ) )  |  E. w  e.  NN0  E. x  e.  NN0  [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph }
7 rexfrabdioph.1 . . . . . . 7  |-  M  =  ( N  +  1 )
8 nn0p1nn 10836 . . . . . . 7  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  NN )
97, 8syl5eqel 2559 . . . . . 6  |-  ( N  e.  NN0  ->  M  e.  NN )
109nnnn0d 10853 . . . . 5  |-  ( N  e.  NN0  ->  M  e. 
NN0 )
1110adantr 465 . . . 4  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  M  e.  NN0 )
12 sbcrot3 30555 . . . . . . . . . . 11  |-  ( [. ( a `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph  <->  [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. [. ( a `  M
)  /  v ]. ph )
1312sbcbii 3391 . . . . . . . . . 10  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a `  M )  /  v ]. ph )
14 sbcrot3 30555 . . . . . . . . . 10  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
a `  M )  /  v ]. ph  <->  [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph )
1513, 14bitri 249 . . . . . . . . 9  |-  ( [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph )
1615sbcbii 3391 . . . . . . . 8  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph )
17 reseq1 5267 . . . . . . . . . 10  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a  |`  ( 1 ... N ) )  =  ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) ) )
1817sbccomieg 30557 . . . . . . . . 9  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) )  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph )
19 fzssp1 11727 . . . . . . . . . . . 12  |-  ( 1 ... N )  C_  ( 1 ... ( N  +  1 ) )
207oveq2i 6296 . . . . . . . . . . . 12  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
2119, 20sseqtr4i 3537 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  ( 1 ... M
)
22 resabs1 5302 . . . . . . . . . . 11  |-  ( ( 1 ... N ) 
C_  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) ) )
23 dfsbcq 3333 . . . . . . . . . . 11  |-  ( ( ( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  =  ( t  |`  ( 1 ... N ) )  ->  ( [. (
( t  |`  (
1 ... M ) )  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
2421, 22, 23mp2b 10 . . . . . . . . . 10  |-  ( [. ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( a `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph )
25 vex 3116 . . . . . . . . . . . . . 14  |-  t  e. 
_V
2625resex 5317 . . . . . . . . . . . . 13  |-  ( t  |`  ( 1 ... M
) )  e.  _V
27 fveq1 5865 . . . . . . . . . . . . . 14  |-  ( a  =  ( t  |`  ( 1 ... M
) )  ->  (
a `  M )  =  ( ( t  |`  ( 1 ... M
) ) `  M
) )
2827sbcco3g 3843 . . . . . . . . . . . . 13  |-  ( ( t  |`  ( 1 ... M ) )  e.  _V  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph ) )
2926, 28ax-mp 5 . . . . . . . . . . . 12  |-  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( ( t  |`  ( 1 ... M ) ) `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph )
30 elfz1end 11716 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  <->  M  e.  ( 1 ... M
) )
319, 30sylib 196 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  M  e.  ( 1 ... M
) )
32 fvres 5880 . . . . . . . . . . . . 13  |-  ( M  e.  ( 1 ... M )  ->  (
( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
) )
33 dfsbcq 3333 . . . . . . . . . . . . 13  |-  ( ( ( t  |`  (
1 ... M ) ) `
 M )  =  ( t `  M
)  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph ) )
3431, 32, 333syl 20 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... M
) ) `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph  <->  [. ( t `  M
)  /  v ]. [. ( t `  L
)  /  w ]. [. ( t `  K
)  /  x ]. ph ) )
3529, 34syl5bb 257 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a `  M )  /  v ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. ph  <->  [. ( t `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph ) )
3635sbcbidv 3390 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
3724, 36syl5bb 257 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( [. ( ( t  |`  ( 1 ... M
) )  |`  (
1 ... N ) )  /  u ]. [. (
t  |`  ( 1 ... M ) )  / 
a ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
3818, 37syl5bb 257 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. [. ( t `
 L )  /  w ]. [. ( t `
 K )  /  x ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
3916, 38syl5bbr 259 . . . . . . 7  |-  ( N  e.  NN0  ->  ( [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph  <->  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph ) )
4039rabbidv 3105 . . . . . 6  |-  ( N  e.  NN0  ->  { t  e.  ( NN0  ^m  ( 1 ... K
) )  |  [. ( t  |`  (
1 ... M ) )  /  a ]. [. (
t `  L )  /  w ]. [. (
t `  K )  /  x ]. [. (
a  |`  ( 1 ... N ) )  /  u ]. [. ( a `
 M )  / 
v ]. ph }  =  { t  e.  ( NN0  ^m  ( 1 ... K ) )  |  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph } )
4140eleq1d 2536 . . . . 5  |-  ( N  e.  NN0  ->  ( { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  K )  <->  { t  e.  ( NN0  ^m  (
1 ... K ) )  |  [. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) ) )
4241biimpar 485 . . . 4  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { t  e.  ( NN0  ^m  ( 1 ... K ) )  |  [. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  K ) )
43 rexfrabdioph.2 . . . . 5  |-  L  =  ( M  +  1 )
44 rexfrabdioph.3 . . . . 5  |-  K  =  ( L  +  1 )
4543, 442rexfrabdioph 30560 . . . 4  |-  ( ( M  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... M
) )  /  a ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. ph }  e.  (Dioph `  K ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  E. w  e. 
NN0  E. x  e.  NN0  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. ph }  e.  (Dioph `  M )
)
4611, 42, 45syl2anc 661 . . 3  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  E. w  e. 
NN0  E. x  e.  NN0  [. ( a  |`  (
1 ... N ) )  /  u ]. [. (
a `  M )  /  v ]. ph }  e.  (Dioph `  M )
)
476, 46syl5eqel 2559 . 2  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { a  e.  ( NN0  ^m  ( 1 ... M ) )  |  [. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  M )
)
487rexfrabdioph 30559 . 2  |-  ( ( N  e.  NN0  /\  { a  e.  ( NN0 
^m  ( 1 ... M ) )  | 
[. ( a  |`  ( 1 ... N
) )  /  u ]. [. ( a `  M )  /  v ]. E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  M )
)  ->  { u  e.  ( NN0  ^m  (
1 ... N ) )  |  E. v  e. 
NN0  E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  N )
)
4947, 48syldan 470 1  |-  ( ( N  e.  NN0  /\  { t  e.  ( NN0 
^m  ( 1 ... K ) )  | 
[. ( t  |`  ( 1 ... N
) )  /  u ]. [. ( t `  M )  /  v ]. [. ( t `  L )  /  w ]. [. ( t `  K )  /  x ]. ph }  e.  (Dioph `  K ) )  ->  { u  e.  ( NN0  ^m  ( 1 ... N ) )  |  E. v  e.  NN0  E. w  e.  NN0  E. x  e.  NN0  ph }  e.  (Dioph `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113   [.wsbc 3331    C_ wss 3476    |` cres 5001   ` cfv 5588  (class class class)co 6285    ^m cmap 7421   1c1 9494    + caddc 9496   NNcn 10537   NN0cn0 10796   ...cfz 11673  Diophcdioph 30519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-hash 12375  df-mzpcl 30486  df-mzp 30487  df-dioph 30520
This theorem is referenced by:  expdiophlem2  30795
  Copyright terms: Public domain W3C validator