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Definition df-pell1234qr 29197
Description: Define the general solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
df-pell1234qr  |- Pell1234QR  =  ( x  e.  ( NN 
\NN )  |->  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  x
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 ) } )
Distinct variable group:    x, y, z, w

Detailed syntax breakdown of Definition df-pell1234qr
StepHypRef Expression
1 cpell1234qr 29191 . 2  class Pell1234QR
2 vx . . 3  setvar  x
3 cn 10334 . . . 4  class  NN
4 csquarenn 29189 . . . 4  classNN
53, 4cdif 3337 . . 3  class  ( NN 
\NN )
6 vy . . . . . . . . 9  setvar  y
76cv 1368 . . . . . . . 8  class  y
8 vz . . . . . . . . . 10  setvar  z
98cv 1368 . . . . . . . . 9  class  z
102cv 1368 . . . . . . . . . . 11  class  x
11 csqr 12734 . . . . . . . . . . 11  class  sqr
1210, 11cfv 5430 . . . . . . . . . 10  class  ( sqr `  x )
13 vw . . . . . . . . . . 11  setvar  w
1413cv 1368 . . . . . . . . . 10  class  w
15 cmul 9299 . . . . . . . . . 10  class  x.
1612, 14, 15co 6103 . . . . . . . . 9  class  ( ( sqr `  x )  x.  w )
17 caddc 9297 . . . . . . . . 9  class  +
189, 16, 17co 6103 . . . . . . . 8  class  ( z  +  ( ( sqr `  x )  x.  w
) )
197, 18wceq 1369 . . . . . . 7  wff  y  =  ( z  +  ( ( sqr `  x
)  x.  w ) )
20 c2 10383 . . . . . . . . . 10  class  2
21 cexp 11877 . . . . . . . . . 10  class  ^
229, 20, 21co 6103 . . . . . . . . 9  class  ( z ^ 2 )
2314, 20, 21co 6103 . . . . . . . . . 10  class  ( w ^ 2 )
2410, 23, 15co 6103 . . . . . . . . 9  class  ( x  x.  ( w ^
2 ) )
25 cmin 9607 . . . . . . . . 9  class  -
2622, 24, 25co 6103 . . . . . . . 8  class  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )
27 c1 9295 . . . . . . . 8  class  1
2826, 27wceq 1369 . . . . . . 7  wff  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1
2919, 28wa 369 . . . . . 6  wff  ( y  =  ( z  +  ( ( sqr `  x
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 )
30 cz 10658 . . . . . 6  class  ZZ
3129, 13, 30wrex 2728 . . . . 5  wff  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  x
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 )
3231, 8, 30wrex 2728 . . . 4  wff  E. z  e.  ZZ  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  x
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 )
33 cr 9293 . . . 4  class  RR
3432, 6, 33crab 2731 . . 3  class  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  x
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 ) }
352, 5, 34cmpt 4362 . 2  class  ( x  e.  ( NN  \NN )  |->  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  (
y  =  ( z  +  ( ( sqr `  x )  x.  w
) )  /\  (
( z ^ 2 )  -  ( x  x.  ( w ^
2 ) ) )  =  1 ) } )
361, 35wceq 1369 1  wff Pell1234QR  =  ( x  e.  ( NN 
\NN )  |->  { y  e.  RR  |  E. z  e.  ZZ  E. w  e.  ZZ  ( y  =  ( z  +  ( ( sqr `  x
)  x.  w ) )  /\  ( ( z ^ 2 )  -  ( x  x.  ( w ^ 2 ) ) )  =  1 ) } )
Colors of variables: wff setvar class
This definition is referenced by:  pell1234qrval  29203
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