Metamath Proof Explorer

Theorem eldioph

Description: Condition for a set to be Diophantine (unpacking existential quantifier). (Contributed by Stefan O'Rear, 5-Oct-2014)

Ref Expression
Assertion eldioph 
|- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. ( mzPoly ` ( 1 ... K ) ) ) -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. ( Dioph ` N ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. ( mzPoly ` ( 1 ... K ) ) ) -> N e. NN0 )
2 simp2 
 |-  ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. ( mzPoly ` ( 1 ... K ) ) ) -> K e. ( ZZ>= ` N ) )
3 simp3 
 |-  ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. ( mzPoly ` ( 1 ... K ) ) ) -> P e. ( mzPoly ` ( 1 ... K ) ) )
4 eqidd 
 |-  ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. ( mzPoly ` ( 1 ... K ) ) ) -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } )
5 fveq1 
 |-  ( p = P -> ( p ` u ) = ( P ` u ) )
6 5 eqeq1d 
 |-  ( p = P -> ( ( p ` u ) = 0 <-> ( P ` u ) = 0 ) )
7 6 anbi2d 
 |-  ( p = P -> ( ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) <-> ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) ) )
8 7 rexbidv 
 |-  ( p = P -> ( E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) <-> E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) ) )
9 8 abbidv 
 |-  ( p = P -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } )
10 9 rspceeqv 
 |-  ( ( P e. ( mzPoly ` ( 1 ... K ) ) /\ { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } ) -> E. p e. ( mzPoly ` ( 1 ... K ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
11 3 4 10 syl2anc 
 |-  ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. ( mzPoly ` ( 1 ... K ) ) ) -> E. p e. ( mzPoly ` ( 1 ... K ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
12 oveq2 
 |-  ( k = K -> ( 1 ... k ) = ( 1 ... K ) )
13 12 fveq2d 
 |-  ( k = K -> ( mzPoly ` ( 1 ... k ) ) = ( mzPoly ` ( 1 ... K ) ) )
14 12 oveq2d 
 |-  ( k = K -> ( NN0 ^m ( 1 ... k ) ) = ( NN0 ^m ( 1 ... K ) ) )
15 14 rexeqdv 
 |-  ( k = K -> ( E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) <-> E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) ) )
16 15 abbidv 
 |-  ( k = K -> { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
17 16 eqeq2d 
 |-  ( k = K -> ( { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } <-> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
18 13 17 rexeqbidv 
 |-  ( k = K -> ( E. p e. ( mzPoly ` ( 1 ... k ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } <-> E. p e. ( mzPoly ` ( 1 ... K ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
19 18 rspcev 
 |-  ( ( K e. ( ZZ>= ` N ) /\ E. p e. ( mzPoly ` ( 1 ... K ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) -> E. k e. ( ZZ>= ` N ) E. p e. ( mzPoly ` ( 1 ... k ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
20 2 11 19 syl2anc 
 |-  ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. ( mzPoly ` ( 1 ... K ) ) ) -> E. k e. ( ZZ>= ` N ) E. p e. ( mzPoly ` ( 1 ... k ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } )
21 eldiophb 
 |-  ( { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. ( Dioph ` N ) <-> ( N e. NN0 /\ E. k e. ( ZZ>= ` N ) E. p e. ( mzPoly ` ( 1 ... k ) ) { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } = { t | E. u e. ( NN0 ^m ( 1 ... k ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( p ` u ) = 0 ) } ) )
22 1 20 21 sylanbrc 
 |-  ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) /\ P e. ( mzPoly ` ( 1 ... K ) ) ) -> { t | E. u e. ( NN0 ^m ( 1 ... K ) ) ( t = ( u |` ( 1 ... N ) ) /\ ( P ` u ) = 0 ) } e. ( Dioph ` N ) )