Step |
Hyp |
Ref |
Expression |
1 |
|
2sqnn |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) |
2 |
|
simpll |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 ∈ ℕ ) |
3 |
2
|
adantl |
⊢ ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → 𝑥 ∈ ℕ ) |
4 |
|
breq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 ≤ 𝑏 ↔ 𝑥 ≤ 𝑏 ) ) |
5 |
|
oveq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) |
6 |
5
|
oveq1d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) ) |
7 |
6
|
eqeq1d |
⊢ ( 𝑎 = 𝑥 → ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ↔ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
8 |
4 7
|
anbi12d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
9 |
8
|
reubidv |
⊢ ( 𝑎 = 𝑥 → ( ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
10 |
9
|
adantl |
⊢ ( ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ∧ 𝑎 = 𝑥 ) → ( ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
11 |
|
simpr |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → 𝑦 ∈ ℕ ) |
12 |
11
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) → 𝑦 ∈ ℕ ) |
13 |
|
breq2 |
⊢ ( 𝑏 = 𝑦 → ( 𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑦 ) ) |
14 |
|
oveq1 |
⊢ ( 𝑏 = 𝑦 → ( 𝑏 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) |
15 |
14
|
oveq2d |
⊢ ( 𝑏 = 𝑦 → ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) |
16 |
15
|
eqeq1d |
⊢ ( 𝑏 = 𝑦 → ( ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
17 |
13 16
|
anbi12d |
⊢ ( 𝑏 = 𝑦 → ( ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
18 |
|
equequ1 |
⊢ ( 𝑏 = 𝑦 → ( 𝑏 = 𝑐 ↔ 𝑦 = 𝑐 ) ) |
19 |
18
|
imbi2d |
⊢ ( 𝑏 = 𝑦 → ( ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ↔ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑦 = 𝑐 ) ) ) |
20 |
19
|
ralbidv |
⊢ ( 𝑏 = 𝑦 → ( ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ↔ ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑦 = 𝑐 ) ) ) |
21 |
17 20
|
anbi12d |
⊢ ( 𝑏 = 𝑦 → ( ( ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ) ↔ ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑦 = 𝑐 ) ) ) ) |
22 |
21
|
adantl |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑏 = 𝑦 ) → ( ( ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ) ↔ ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑦 = 𝑐 ) ) ) ) |
23 |
|
simpr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) → 𝑥 ≤ 𝑦 ) |
24 |
|
eqidd |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) → ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) |
25 |
|
nnre |
⊢ ( 𝑐 ∈ ℕ → 𝑐 ∈ ℝ ) |
26 |
25
|
resqcld |
⊢ ( 𝑐 ∈ ℕ → ( 𝑐 ↑ 2 ) ∈ ℝ ) |
27 |
26
|
adantl |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) → ( 𝑐 ↑ 2 ) ∈ ℝ ) |
28 |
|
nnre |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℝ ) |
29 |
28
|
resqcld |
⊢ ( 𝑦 ∈ ℕ → ( 𝑦 ↑ 2 ) ∈ ℝ ) |
30 |
29
|
adantl |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑦 ↑ 2 ) ∈ ℝ ) |
31 |
30
|
ad2antrr |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) → ( 𝑦 ↑ 2 ) ∈ ℝ ) |
32 |
|
nnre |
⊢ ( 𝑥 ∈ ℕ → 𝑥 ∈ ℝ ) |
33 |
32
|
resqcld |
⊢ ( 𝑥 ∈ ℕ → ( 𝑥 ↑ 2 ) ∈ ℝ ) |
34 |
33
|
adantr |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑥 ↑ 2 ) ∈ ℝ ) |
35 |
34
|
ad2antrr |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) → ( 𝑥 ↑ 2 ) ∈ ℝ ) |
36 |
|
readdcan |
⊢ ( ( ( 𝑐 ↑ 2 ) ∈ ℝ ∧ ( 𝑦 ↑ 2 ) ∈ ℝ ∧ ( 𝑥 ↑ 2 ) ∈ ℝ ) → ( ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) ) |
37 |
27 31 35 36
|
syl3anc |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) → ( ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) ) |
38 |
28
|
ad4antlr |
⊢ ( ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) → 𝑦 ∈ ℝ ) |
39 |
25
|
ad2antlr |
⊢ ( ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) → 𝑐 ∈ ℝ ) |
40 |
|
nnnn0 |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℕ0 ) |
41 |
40
|
nn0ge0d |
⊢ ( 𝑦 ∈ ℕ → 0 ≤ 𝑦 ) |
42 |
41
|
ad4antlr |
⊢ ( ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) → 0 ≤ 𝑦 ) |
43 |
|
nnnn0 |
⊢ ( 𝑐 ∈ ℕ → 𝑐 ∈ ℕ0 ) |
44 |
43
|
nn0ge0d |
⊢ ( 𝑐 ∈ ℕ → 0 ≤ 𝑐 ) |
45 |
44
|
ad2antlr |
⊢ ( ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) → 0 ≤ 𝑐 ) |
46 |
|
simpr |
⊢ ( ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) → ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) |
47 |
46
|
eqcomd |
⊢ ( ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) → ( 𝑦 ↑ 2 ) = ( 𝑐 ↑ 2 ) ) |
48 |
38 39 42 45 47
|
sq11d |
⊢ ( ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) → 𝑦 = 𝑐 ) |
49 |
48
|
ex |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) → ( ( 𝑐 ↑ 2 ) = ( 𝑦 ↑ 2 ) → 𝑦 = 𝑐 ) ) |
50 |
37 49
|
sylbid |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) → ( ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → 𝑦 = 𝑐 ) ) |
51 |
50
|
adantld |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) ∧ 𝑐 ∈ ℕ ) → ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑦 = 𝑐 ) ) |
52 |
51
|
ralrimiva |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) → ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑦 = 𝑐 ) ) |
53 |
23 24 52
|
jca31 |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) → ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑦 = 𝑐 ) ) ) |
54 |
12 22 53
|
rspcedvd |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) → ∃ 𝑏 ∈ ℕ ( ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ) ) |
55 |
|
breq2 |
⊢ ( 𝑏 = 𝑐 → ( 𝑥 ≤ 𝑏 ↔ 𝑥 ≤ 𝑐 ) ) |
56 |
|
oveq1 |
⊢ ( 𝑏 = 𝑐 → ( 𝑏 ↑ 2 ) = ( 𝑐 ↑ 2 ) ) |
57 |
56
|
oveq2d |
⊢ ( 𝑏 = 𝑐 → ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) ) |
58 |
57
|
eqeq1d |
⊢ ( 𝑏 = 𝑐 → ( ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
59 |
55 58
|
anbi12d |
⊢ ( 𝑏 = 𝑐 → ( ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
60 |
59
|
reu8 |
⊢ ( ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ∃ 𝑏 ∈ ℕ ( ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑥 ≤ 𝑐 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ) ) |
61 |
54 60
|
sylibr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑥 ≤ 𝑦 ) → ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
62 |
61
|
ex |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑥 ≤ 𝑦 → ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
63 |
62
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → ( 𝑥 ≤ 𝑦 → ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
64 |
63
|
impcom |
⊢ ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
65 |
|
eqeq2 |
⊢ ( 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ( ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ↔ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
66 |
65
|
anbi2d |
⊢ ( 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ( ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
67 |
66
|
reubidv |
⊢ ( 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ( ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
68 |
67
|
adantl |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → ( ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
69 |
68
|
adantl |
⊢ ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → ( ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
70 |
64 69
|
mpbird |
⊢ ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → ∃! 𝑏 ∈ ℕ ( 𝑥 ≤ 𝑏 ∧ ( ( 𝑥 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
71 |
3 10 70
|
rspcedvd |
⊢ ( ( 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → ∃ 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
72 |
11
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑦 ∈ ℕ ) |
73 |
72
|
adantl |
⊢ ( ( ¬ 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → 𝑦 ∈ ℕ ) |
74 |
|
breq1 |
⊢ ( 𝑎 = 𝑦 → ( 𝑎 ≤ 𝑏 ↔ 𝑦 ≤ 𝑏 ) ) |
75 |
|
oveq1 |
⊢ ( 𝑎 = 𝑦 → ( 𝑎 ↑ 2 ) = ( 𝑦 ↑ 2 ) ) |
76 |
75
|
oveq1d |
⊢ ( 𝑎 = 𝑦 → ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) ) |
77 |
76
|
eqeq1d |
⊢ ( 𝑎 = 𝑦 → ( ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ↔ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
78 |
74 77
|
anbi12d |
⊢ ( 𝑎 = 𝑦 → ( ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
79 |
78
|
reubidv |
⊢ ( 𝑎 = 𝑦 → ( ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
80 |
79
|
adantl |
⊢ ( ( ( ¬ 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ∧ 𝑎 = 𝑦 ) → ( ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
81 |
|
simpll |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ ¬ 𝑥 ≤ 𝑦 ) → 𝑥 ∈ ℕ ) |
82 |
|
breq2 |
⊢ ( 𝑏 = 𝑥 → ( 𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑥 ) ) |
83 |
|
oveq1 |
⊢ ( 𝑏 = 𝑥 → ( 𝑏 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) |
84 |
83
|
oveq2d |
⊢ ( 𝑏 = 𝑥 → ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) ) |
85 |
84
|
eqeq1d |
⊢ ( 𝑏 = 𝑥 → ( ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
86 |
82 85
|
anbi12d |
⊢ ( 𝑏 = 𝑥 → ( ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ( 𝑦 ≤ 𝑥 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
87 |
|
equequ1 |
⊢ ( 𝑏 = 𝑥 → ( 𝑏 = 𝑐 ↔ 𝑥 = 𝑐 ) ) |
88 |
87
|
imbi2d |
⊢ ( 𝑏 = 𝑥 → ( ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ↔ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 = 𝑐 ) ) ) |
89 |
88
|
ralbidv |
⊢ ( 𝑏 = 𝑥 → ( ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ↔ ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 = 𝑐 ) ) ) |
90 |
86 89
|
anbi12d |
⊢ ( 𝑏 = 𝑥 → ( ( ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ) ↔ ( ( 𝑦 ≤ 𝑥 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 = 𝑐 ) ) ) ) |
91 |
90
|
adantl |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ ¬ 𝑥 ≤ 𝑦 ) ∧ 𝑏 = 𝑥 ) → ( ( ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ) ↔ ( ( 𝑦 ≤ 𝑥 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 = 𝑐 ) ) ) ) |
92 |
|
ltnle |
⊢ ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦 ) ) |
93 |
28 32 92
|
syl2anr |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑦 < 𝑥 ↔ ¬ 𝑥 ≤ 𝑦 ) ) |
94 |
28
|
ad2antlr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 𝑥 ) → 𝑦 ∈ ℝ ) |
95 |
32
|
ad2antrr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 𝑥 ) → 𝑥 ∈ ℝ ) |
96 |
|
simpr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 𝑥 ) → 𝑦 < 𝑥 ) |
97 |
94 95 96
|
ltled |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑦 < 𝑥 ) → 𝑦 ≤ 𝑥 ) |
98 |
97
|
ex |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑦 < 𝑥 → 𝑦 ≤ 𝑥 ) ) |
99 |
93 98
|
sylbird |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( ¬ 𝑥 ≤ 𝑦 → 𝑦 ≤ 𝑥 ) ) |
100 |
99
|
imp |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ ¬ 𝑥 ≤ 𝑦 ) → 𝑦 ≤ 𝑥 ) |
101 |
29
|
recnd |
⊢ ( 𝑦 ∈ ℕ → ( 𝑦 ↑ 2 ) ∈ ℂ ) |
102 |
101
|
adantl |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑦 ↑ 2 ) ∈ ℂ ) |
103 |
33
|
recnd |
⊢ ( 𝑥 ∈ ℕ → ( 𝑥 ↑ 2 ) ∈ ℂ ) |
104 |
103
|
adantr |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑥 ↑ 2 ) ∈ ℂ ) |
105 |
102 104
|
addcomd |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) |
106 |
105
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ ¬ 𝑥 ≤ 𝑦 ) → ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) |
107 |
34
|
recnd |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑥 ↑ 2 ) ∈ ℂ ) |
108 |
107
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( 𝑥 ↑ 2 ) ∈ ℂ ) |
109 |
30
|
recnd |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑦 ↑ 2 ) ∈ ℂ ) |
110 |
109
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( 𝑦 ↑ 2 ) ∈ ℂ ) |
111 |
108 110
|
addcomd |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) ) |
112 |
111
|
eqeq2d |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) ) ) |
113 |
26
|
adantl |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( 𝑐 ↑ 2 ) ∈ ℝ ) |
114 |
33
|
ad2antrr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( 𝑥 ↑ 2 ) ∈ ℝ ) |
115 |
29
|
ad2antlr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( 𝑦 ↑ 2 ) ∈ ℝ ) |
116 |
|
readdcan |
⊢ ( ( ( 𝑐 ↑ 2 ) ∈ ℝ ∧ ( 𝑥 ↑ 2 ) ∈ ℝ ∧ ( 𝑦 ↑ 2 ) ∈ ℝ ) → ( ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) ↔ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) ) |
117 |
113 114 115 116
|
syl3anc |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) ↔ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) ) |
118 |
112 117
|
bitrd |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) ) |
119 |
25
|
ad2antlr |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) → 𝑐 ∈ ℝ ) |
120 |
32
|
adantr |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → 𝑥 ∈ ℝ ) |
121 |
120
|
ad2antrr |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) → 𝑥 ∈ ℝ ) |
122 |
44
|
ad2antlr |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) → 0 ≤ 𝑐 ) |
123 |
|
nnnn0 |
⊢ ( 𝑥 ∈ ℕ → 𝑥 ∈ ℕ0 ) |
124 |
123
|
nn0ge0d |
⊢ ( 𝑥 ∈ ℕ → 0 ≤ 𝑥 ) |
125 |
124
|
adantr |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → 0 ≤ 𝑥 ) |
126 |
125
|
ad2antrr |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) → 0 ≤ 𝑥 ) |
127 |
|
simpr |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) → ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) |
128 |
119 121 122 126 127
|
sq11d |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) → 𝑐 = 𝑥 ) |
129 |
128
|
eqcomd |
⊢ ( ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) ∧ ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) ) → 𝑥 = 𝑐 ) |
130 |
129
|
ex |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( ( 𝑐 ↑ 2 ) = ( 𝑥 ↑ 2 ) → 𝑥 = 𝑐 ) ) |
131 |
118 130
|
sylbid |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → 𝑥 = 𝑐 ) ) |
132 |
131
|
adantld |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑐 ∈ ℕ ) → ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 = 𝑐 ) ) |
133 |
132
|
ralrimiva |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 = 𝑐 ) ) |
134 |
133
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ ¬ 𝑥 ≤ 𝑦 ) → ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 = 𝑐 ) ) |
135 |
100 106 134
|
jca31 |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ ¬ 𝑥 ≤ 𝑦 ) → ( ( 𝑦 ≤ 𝑥 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑥 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑥 = 𝑐 ) ) ) |
136 |
81 91 135
|
rspcedvd |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ ¬ 𝑥 ≤ 𝑦 ) → ∃ 𝑏 ∈ ℕ ( ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ) ) |
137 |
|
breq2 |
⊢ ( 𝑏 = 𝑐 → ( 𝑦 ≤ 𝑏 ↔ 𝑦 ≤ 𝑐 ) ) |
138 |
56
|
oveq2d |
⊢ ( 𝑏 = 𝑐 → ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) ) |
139 |
138
|
eqeq1d |
⊢ ( 𝑏 = 𝑐 → ( ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
140 |
137 139
|
anbi12d |
⊢ ( 𝑏 = 𝑐 → ( ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
141 |
140
|
reu8 |
⊢ ( ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ∃ 𝑏 ∈ ℕ ( ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ∧ ∀ 𝑐 ∈ ℕ ( ( 𝑦 ≤ 𝑐 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑐 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → 𝑏 = 𝑐 ) ) ) |
142 |
136 141
|
sylibr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ ¬ 𝑥 ≤ 𝑦 ) → ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
143 |
142
|
ex |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( ¬ 𝑥 ≤ 𝑦 → ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
144 |
143
|
adantr |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → ( ¬ 𝑥 ≤ 𝑦 → ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
145 |
144
|
impcom |
⊢ ( ( ¬ 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
146 |
|
eqeq2 |
⊢ ( 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ( ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ↔ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
147 |
146
|
anbi2d |
⊢ ( 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ( ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
148 |
147
|
reubidv |
⊢ ( 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ( ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
149 |
148
|
adantl |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → ( ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
150 |
149
|
adantl |
⊢ ( ( ¬ 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → ( ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
151 |
145 150
|
mpbird |
⊢ ( ( ¬ 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → ∃! 𝑏 ∈ ℕ ( 𝑦 ≤ 𝑏 ∧ ( ( 𝑦 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
152 |
73 80 151
|
rspcedvd |
⊢ ( ( ¬ 𝑥 ≤ 𝑦 ∧ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) → ∃ 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
153 |
71 152
|
pm2.61ian |
⊢ ( ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ∧ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) → ∃ 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
154 |
153
|
ex |
⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ∃ 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
155 |
154
|
adantl |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) ) → ( 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ∃ 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
156 |
155
|
rexlimdvva |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ 𝑃 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) → ∃ 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
157 |
1 156
|
mpd |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ∃ 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
158 |
|
reurex |
⊢ ( ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
159 |
158
|
a1i |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ 𝑎 ∈ ℕ ) → ( ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
160 |
159
|
ralrimiva |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ∀ 𝑎 ∈ ℕ ( ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
161 |
|
2sqmo |
⊢ ( 𝑃 ∈ ℙ → ∃* 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
162 |
|
nnssnn0 |
⊢ ℕ ⊆ ℕ0 |
163 |
|
nfcv |
⊢ Ⅎ 𝑎 ℕ |
164 |
|
nfcv |
⊢ Ⅎ 𝑎 ℕ0 |
165 |
163 164
|
ssrmof |
⊢ ( ℕ ⊆ ℕ0 → ( ∃* 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃* 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ0 ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
166 |
162 165
|
ax-mp |
⊢ ( ∃* 𝑎 ∈ ℕ0 ∃ 𝑏 ∈ ℕ0 ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃* 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ0 ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
167 |
|
ssrexv |
⊢ ( ℕ ⊆ ℕ0 → ( ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃ 𝑏 ∈ ℕ0 ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
168 |
162 167
|
ax-mp |
⊢ ( ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃ 𝑏 ∈ ℕ0 ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
169 |
168
|
rmoimi |
⊢ ( ∃* 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ0 ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃* 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
170 |
161 166 169
|
3syl |
⊢ ( 𝑃 ∈ ℙ → ∃* 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
171 |
170
|
adantr |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ∃* 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
172 |
|
rmoim |
⊢ ( ∀ 𝑎 ∈ ℕ ( ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) → ( ∃* 𝑎 ∈ ℕ ∃ 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) → ∃* 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
173 |
160 171 172
|
sylc |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ∃* 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |
174 |
|
reu5 |
⊢ ( ∃! 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ↔ ( ∃ 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ∧ ∃* 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) ) |
175 |
157 173 174
|
sylanbrc |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ∃! 𝑎 ∈ ℕ ∃! 𝑏 ∈ ℕ ( 𝑎 ≤ 𝑏 ∧ ( ( 𝑎 ↑ 2 ) + ( 𝑏 ↑ 2 ) ) = 𝑃 ) ) |