Step |
Hyp |
Ref |
Expression |
1 |
|
2sqreultlem |
|- ( ( P e. Prime /\ ( P mod 4 ) = 1 ) -> E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) |
2 |
1
|
ex |
|- ( P e. Prime -> ( ( P mod 4 ) = 1 -> E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) ) |
3 |
|
2reu2rex |
|- ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> E. a e. NN0 E. b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) |
4 |
|
elsni |
|- ( P e. { 2 } -> P = 2 ) |
5 |
|
eqeq2 |
|- ( P = 2 -> ( ( ( a ^ 2 ) + ( b ^ 2 ) ) = P <-> ( ( a ^ 2 ) + ( b ^ 2 ) ) = 2 ) ) |
6 |
5
|
anbi2d |
|- ( P = 2 -> ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = 2 ) ) ) |
7 |
6
|
adantl |
|- ( ( ( a e. NN0 /\ b e. NN0 ) /\ P = 2 ) -> ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) <-> ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = 2 ) ) ) |
8 |
|
2sq2 |
|- ( ( a e. NN0 /\ b e. NN0 ) -> ( ( ( a ^ 2 ) + ( b ^ 2 ) ) = 2 <-> ( a = 1 /\ b = 1 ) ) ) |
9 |
|
breq12 |
|- ( ( a = 1 /\ b = 1 ) -> ( a < b <-> 1 < 1 ) ) |
10 |
|
1re |
|- 1 e. RR |
11 |
10
|
ltnri |
|- -. 1 < 1 |
12 |
11
|
pm2.21i |
|- ( 1 < 1 -> ( P mod 4 ) = 1 ) |
13 |
9 12
|
syl6bi |
|- ( ( a = 1 /\ b = 1 ) -> ( a < b -> ( P mod 4 ) = 1 ) ) |
14 |
8 13
|
syl6bi |
|- ( ( a e. NN0 /\ b e. NN0 ) -> ( ( ( a ^ 2 ) + ( b ^ 2 ) ) = 2 -> ( a < b -> ( P mod 4 ) = 1 ) ) ) |
15 |
14
|
impcomd |
|- ( ( a e. NN0 /\ b e. NN0 ) -> ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = 2 ) -> ( P mod 4 ) = 1 ) ) |
16 |
15
|
adantr |
|- ( ( ( a e. NN0 /\ b e. NN0 ) /\ P = 2 ) -> ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = 2 ) -> ( P mod 4 ) = 1 ) ) |
17 |
7 16
|
sylbid |
|- ( ( ( a e. NN0 /\ b e. NN0 ) /\ P = 2 ) -> ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P mod 4 ) = 1 ) ) |
18 |
17
|
ex |
|- ( ( a e. NN0 /\ b e. NN0 ) -> ( P = 2 -> ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P mod 4 ) = 1 ) ) ) |
19 |
18
|
com23 |
|- ( ( a e. NN0 /\ b e. NN0 ) -> ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P = 2 -> ( P mod 4 ) = 1 ) ) ) |
20 |
19
|
rexlimivv |
|- ( E. a e. NN0 E. b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P = 2 -> ( P mod 4 ) = 1 ) ) |
21 |
3 4 20
|
syl2imc |
|- ( P e. { 2 } -> ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P mod 4 ) = 1 ) ) |
22 |
21
|
a1d |
|- ( P e. { 2 } -> ( P e. Prime -> ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P mod 4 ) = 1 ) ) ) |
23 |
|
eldif |
|- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ -. P e. { 2 } ) ) |
24 |
|
eldifsnneq |
|- ( P e. ( Prime \ { 2 } ) -> -. P = 2 ) |
25 |
|
nn0ssz |
|- NN0 C_ ZZ |
26 |
|
id |
|- ( ( ( a ^ 2 ) + ( b ^ 2 ) ) = P -> ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) |
27 |
26
|
eqcomd |
|- ( ( ( a ^ 2 ) + ( b ^ 2 ) ) = P -> P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
28 |
27
|
adantl |
|- ( ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
29 |
28
|
reximi |
|- ( E. b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> E. b e. NN0 P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
30 |
29
|
reximi |
|- ( E. a e. NN0 E. b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> E. a e. NN0 E. b e. NN0 P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
31 |
|
ssrexv |
|- ( NN0 C_ ZZ -> ( E. b e. NN0 P = ( ( a ^ 2 ) + ( b ^ 2 ) ) -> E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) ) |
32 |
25 31
|
ax-mp |
|- ( E. b e. NN0 P = ( ( a ^ 2 ) + ( b ^ 2 ) ) -> E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
33 |
32
|
reximi |
|- ( E. a e. NN0 E. b e. NN0 P = ( ( a ^ 2 ) + ( b ^ 2 ) ) -> E. a e. NN0 E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
34 |
3 30 33
|
3syl |
|- ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> E. a e. NN0 E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
35 |
|
ssrexv |
|- ( NN0 C_ ZZ -> ( E. a e. NN0 E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) -> E. a e. ZZ E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) ) |
36 |
25 34 35
|
mpsyl |
|- ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> E. a e. ZZ E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
37 |
36
|
adantl |
|- ( ( P e. ( Prime \ { 2 } ) /\ E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) -> E. a e. ZZ E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) ) |
38 |
|
eldifi |
|- ( P e. ( Prime \ { 2 } ) -> P e. Prime ) |
39 |
38
|
adantr |
|- ( ( P e. ( Prime \ { 2 } ) /\ E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) -> P e. Prime ) |
40 |
|
2sqb |
|- ( P e. Prime -> ( E. a e. ZZ E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) <-> ( P = 2 \/ ( P mod 4 ) = 1 ) ) ) |
41 |
39 40
|
syl |
|- ( ( P e. ( Prime \ { 2 } ) /\ E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) -> ( E. a e. ZZ E. b e. ZZ P = ( ( a ^ 2 ) + ( b ^ 2 ) ) <-> ( P = 2 \/ ( P mod 4 ) = 1 ) ) ) |
42 |
37 41
|
mpbid |
|- ( ( P e. ( Prime \ { 2 } ) /\ E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) -> ( P = 2 \/ ( P mod 4 ) = 1 ) ) |
43 |
42
|
ord |
|- ( ( P e. ( Prime \ { 2 } ) /\ E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) -> ( -. P = 2 -> ( P mod 4 ) = 1 ) ) |
44 |
43
|
ex |
|- ( P e. ( Prime \ { 2 } ) -> ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( -. P = 2 -> ( P mod 4 ) = 1 ) ) ) |
45 |
24 44
|
mpid |
|- ( P e. ( Prime \ { 2 } ) -> ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P mod 4 ) = 1 ) ) |
46 |
23 45
|
sylbir |
|- ( ( P e. Prime /\ -. P e. { 2 } ) -> ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P mod 4 ) = 1 ) ) |
47 |
46
|
expcom |
|- ( -. P e. { 2 } -> ( P e. Prime -> ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P mod 4 ) = 1 ) ) ) |
48 |
22 47
|
pm2.61i |
|- ( P e. Prime -> ( E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) -> ( P mod 4 ) = 1 ) ) |
49 |
2 48
|
impbid |
|- ( P e. Prime -> ( ( P mod 4 ) = 1 <-> E! a e. NN0 E! b e. NN0 ( a < b /\ ( ( a ^ 2 ) + ( b ^ 2 ) ) = P ) ) ) |