Step |
Hyp |
Ref |
Expression |
1 |
|
biid |
|- ( ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) |
2 |
1
|
2sqreu |
|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> ( E! a e. NN0 E. b e. NN0 ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) /\ E! b e. NN0 E. a e. NN0 ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) ) |
3 |
|
fveq2 |
|- ( p = <. a , b >. -> ( 1st ` p ) = ( 1st ` <. a , b >. ) ) |
4 |
|
fveq2 |
|- ( p = <. a , b >. -> ( 2nd ` p ) = ( 2nd ` <. a , b >. ) ) |
5 |
3 4
|
breq12d |
|- ( p = <. a , b >. -> ( ( 1st ` p ) <_ ( 2nd ` p ) <-> ( 1st ` <. a , b >. ) <_ ( 2nd ` <. a , b >. ) ) ) |
6 |
|
vex |
|- a e. _V |
7 |
|
vex |
|- b e. _V |
8 |
6 7
|
op1st |
|- ( 1st ` <. a , b >. ) = a |
9 |
6 7
|
op2nd |
|- ( 2nd ` <. a , b >. ) = b |
10 |
8 9
|
breq12i |
|- ( ( 1st ` <. a , b >. ) <_ ( 2nd ` <. a , b >. ) <-> a <_ b ) |
11 |
5 10
|
bitrdi |
|- ( p = <. a , b >. -> ( ( 1st ` p ) <_ ( 2nd ` p ) <-> a <_ b ) ) |
12 |
6 7
|
op1std |
|- ( p = <. a , b >. -> ( 1st ` p ) = a ) |
13 |
12
|
oveq1d |
|- ( p = <. a , b >. -> ( ( 1st ` p ) ^ 2 ) = ( a ^ 2 ) ) |
14 |
6 7
|
op2ndd |
|- ( p = <. a , b >. -> ( 2nd ` p ) = b ) |
15 |
14
|
oveq1d |
|- ( p = <. a , b >. -> ( ( 2nd ` p ) ^ 2 ) = ( b ^ 2 ) ) |
16 |
13 15
|
oveq12d |
|- ( p = <. a , b >. -> ( ( ( 1st ` p ) ^ 2 ) + ( ( 2nd ` p ) ^ 2 ) ) = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
17 |
16
|
eqeq1d |
|- ( p = <. a , b >. -> ( ( ( ( 1st ` p ) ^ 2 ) + ( ( 2nd ` p ) ^ 2 ) ) = P <-> ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) |
18 |
11 17
|
anbi12d |
|- ( p = <. a , b >. -> ( ( ( 1st ` p ) <_ ( 2nd ` p ) /\ ( ( ( 1st ` p ) ^ 2 ) + ( ( 2nd ` p ) ^ 2 ) ) = P ) <-> ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) ) |
19 |
18
|
opreu2reurex |
|- ( E! p e. ( NN0 X. NN0 ) ( ( 1st ` p ) <_ ( 2nd ` p ) /\ ( ( ( 1st ` p ) ^ 2 ) + ( ( 2nd ` p ) ^ 2 ) ) = P ) <-> ( E! a e. NN0 E. b e. NN0 ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) /\ E! b e. NN0 E. a e. NN0 ( a <_ b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) ) |
20 |
2 19
|
sylibr |
|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> E! p e. ( NN0 X. NN0 ) ( ( 1st ` p ) <_ ( 2nd ` p ) /\ ( ( ( 1st ` p ) ^ 2 ) + ( ( 2nd ` p ) ^ 2 ) ) = P ) ) |