Step |
Hyp |
Ref |
Expression |
1 |
|
pell1qrval |
|- ( D e. ( NN \ []NN ) -> ( Pell1QR ` D ) = { a e. RR | E. z e. NN0 E. w e. NN0 ( a = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) } ) |
2 |
1
|
eleq2d |
|- ( D e. ( NN \ []NN ) -> ( A e. ( Pell1QR ` D ) <-> A e. { a e. RR | E. z e. NN0 E. w e. NN0 ( a = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) } ) ) |
3 |
|
eqeq1 |
|- ( a = A -> ( a = ( z + ( ( sqrt ` D ) x. w ) ) <-> A = ( z + ( ( sqrt ` D ) x. w ) ) ) ) |
4 |
3
|
anbi1d |
|- ( a = A -> ( ( a = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) <-> ( A = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) ) ) |
5 |
4
|
2rexbidv |
|- ( a = A -> ( E. z e. NN0 E. w e. NN0 ( a = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) <-> E. z e. NN0 E. w e. NN0 ( A = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) ) ) |
6 |
5
|
elrab |
|- ( A e. { a e. RR | E. z e. NN0 E. w e. NN0 ( a = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) } <-> ( A e. RR /\ E. z e. NN0 E. w e. NN0 ( A = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) ) ) |
7 |
2 6
|
bitrdi |
|- ( D e. ( NN \ []NN ) -> ( A e. ( Pell1QR ` D ) <-> ( A e. RR /\ E. z e. NN0 E. w e. NN0 ( A = ( z + ( ( sqrt ` D ) x. w ) ) /\ ( ( z ^ 2 ) - ( D x. ( w ^ 2 ) ) ) = 1 ) ) ) ) |