Step |
Hyp |
Ref |
Expression |
1 |
|
2sq.1 |
⊢ 𝑆 = ran ( 𝑤 ∈ ℤ[i] ↦ ( ( abs ‘ 𝑤 ) ↑ 2 ) ) |
2 |
|
2sqlem7.2 |
⊢ 𝑌 = { 𝑧 ∣ ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ( ( 𝑥 gcd 𝑦 ) = 1 ∧ 𝑧 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) } |
3 |
|
simpr |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( 𝑃 mod 4 ) = 1 ) |
4 |
|
simpl |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → 𝑃 ∈ ℙ ) |
5 |
|
1ne2 |
⊢ 1 ≠ 2 |
6 |
5
|
necomi |
⊢ 2 ≠ 1 |
7 |
|
oveq1 |
⊢ ( 𝑃 = 2 → ( 𝑃 mod 4 ) = ( 2 mod 4 ) ) |
8 |
|
2re |
⊢ 2 ∈ ℝ |
9 |
|
4re |
⊢ 4 ∈ ℝ |
10 |
|
4pos |
⊢ 0 < 4 |
11 |
9 10
|
elrpii |
⊢ 4 ∈ ℝ+ |
12 |
|
0le2 |
⊢ 0 ≤ 2 |
13 |
|
2lt4 |
⊢ 2 < 4 |
14 |
|
modid |
⊢ ( ( ( 2 ∈ ℝ ∧ 4 ∈ ℝ+ ) ∧ ( 0 ≤ 2 ∧ 2 < 4 ) ) → ( 2 mod 4 ) = 2 ) |
15 |
8 11 12 13 14
|
mp4an |
⊢ ( 2 mod 4 ) = 2 |
16 |
7 15
|
eqtrdi |
⊢ ( 𝑃 = 2 → ( 𝑃 mod 4 ) = 2 ) |
17 |
16
|
neeq1d |
⊢ ( 𝑃 = 2 → ( ( 𝑃 mod 4 ) ≠ 1 ↔ 2 ≠ 1 ) ) |
18 |
6 17
|
mpbiri |
⊢ ( 𝑃 = 2 → ( 𝑃 mod 4 ) ≠ 1 ) |
19 |
18
|
necon2i |
⊢ ( ( 𝑃 mod 4 ) = 1 → 𝑃 ≠ 2 ) |
20 |
3 19
|
syl |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → 𝑃 ≠ 2 ) |
21 |
|
eldifsn |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) ↔ ( 𝑃 ∈ ℙ ∧ 𝑃 ≠ 2 ) ) |
22 |
4 20 21
|
sylanbrc |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) |
23 |
|
m1lgs |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → ( ( - 1 /L 𝑃 ) = 1 ↔ ( 𝑃 mod 4 ) = 1 ) ) |
24 |
22 23
|
syl |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ( - 1 /L 𝑃 ) = 1 ↔ ( 𝑃 mod 4 ) = 1 ) ) |
25 |
3 24
|
mpbird |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( - 1 /L 𝑃 ) = 1 ) |
26 |
|
neg1z |
⊢ - 1 ∈ ℤ |
27 |
|
lgsqr |
⊢ ( ( - 1 ∈ ℤ ∧ 𝑃 ∈ ( ℙ ∖ { 2 } ) ) → ( ( - 1 /L 𝑃 ) = 1 ↔ ( ¬ 𝑃 ∥ - 1 ∧ ∃ 𝑛 ∈ ℤ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) ) |
28 |
26 22 27
|
sylancr |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ( - 1 /L 𝑃 ) = 1 ↔ ( ¬ 𝑃 ∥ - 1 ∧ ∃ 𝑛 ∈ ℤ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) ) |
29 |
25 28
|
mpbid |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ( ¬ 𝑃 ∥ - 1 ∧ ∃ 𝑛 ∈ ℤ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) |
30 |
29
|
simprd |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → ∃ 𝑛 ∈ ℤ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) |
31 |
|
simprl |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → 𝑛 ∈ ℤ ) |
32 |
|
1zzd |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → 1 ∈ ℤ ) |
33 |
|
gcd1 |
⊢ ( 𝑛 ∈ ℤ → ( 𝑛 gcd 1 ) = 1 ) |
34 |
33
|
ad2antrl |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → ( 𝑛 gcd 1 ) = 1 ) |
35 |
|
eqidd |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑛 ↑ 2 ) + 1 ) ) |
36 |
|
oveq1 |
⊢ ( 𝑥 = 𝑛 → ( 𝑥 gcd 𝑦 ) = ( 𝑛 gcd 𝑦 ) ) |
37 |
36
|
eqeq1d |
⊢ ( 𝑥 = 𝑛 → ( ( 𝑥 gcd 𝑦 ) = 1 ↔ ( 𝑛 gcd 𝑦 ) = 1 ) ) |
38 |
|
oveq1 |
⊢ ( 𝑥 = 𝑛 → ( 𝑥 ↑ 2 ) = ( 𝑛 ↑ 2 ) ) |
39 |
38
|
oveq1d |
⊢ ( 𝑥 = 𝑛 → ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑛 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) |
40 |
39
|
eqeq2d |
⊢ ( 𝑥 = 𝑛 → ( ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑛 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
41 |
37 40
|
anbi12d |
⊢ ( 𝑥 = 𝑛 → ( ( ( 𝑥 gcd 𝑦 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ( ( 𝑛 gcd 𝑦 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑛 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
42 |
|
oveq2 |
⊢ ( 𝑦 = 1 → ( 𝑛 gcd 𝑦 ) = ( 𝑛 gcd 1 ) ) |
43 |
42
|
eqeq1d |
⊢ ( 𝑦 = 1 → ( ( 𝑛 gcd 𝑦 ) = 1 ↔ ( 𝑛 gcd 1 ) = 1 ) ) |
44 |
|
oveq1 |
⊢ ( 𝑦 = 1 → ( 𝑦 ↑ 2 ) = ( 1 ↑ 2 ) ) |
45 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
46 |
44 45
|
eqtrdi |
⊢ ( 𝑦 = 1 → ( 𝑦 ↑ 2 ) = 1 ) |
47 |
46
|
oveq2d |
⊢ ( 𝑦 = 1 → ( ( 𝑛 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) = ( ( 𝑛 ↑ 2 ) + 1 ) ) |
48 |
47
|
eqeq2d |
⊢ ( 𝑦 = 1 → ( ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑛 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑛 ↑ 2 ) + 1 ) ) ) |
49 |
43 48
|
anbi12d |
⊢ ( 𝑦 = 1 → ( ( ( 𝑛 gcd 𝑦 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑛 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ( ( 𝑛 gcd 1 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑛 ↑ 2 ) + 1 ) ) ) ) |
50 |
41 49
|
rspc2ev |
⊢ ( ( 𝑛 ∈ ℤ ∧ 1 ∈ ℤ ∧ ( ( 𝑛 gcd 1 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑛 ↑ 2 ) + 1 ) ) ) → ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ( ( 𝑥 gcd 𝑦 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
51 |
31 32 34 35 50
|
syl112anc |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ( ( 𝑥 gcd 𝑦 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
52 |
|
ovex |
⊢ ( ( 𝑛 ↑ 2 ) + 1 ) ∈ V |
53 |
|
eqeq1 |
⊢ ( 𝑧 = ( ( 𝑛 ↑ 2 ) + 1 ) → ( 𝑧 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ↔ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
54 |
53
|
anbi2d |
⊢ ( 𝑧 = ( ( 𝑛 ↑ 2 ) + 1 ) → ( ( ( 𝑥 gcd 𝑦 ) = 1 ∧ 𝑧 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ( ( 𝑥 gcd 𝑦 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
55 |
54
|
2rexbidv |
⊢ ( 𝑧 = ( ( 𝑛 ↑ 2 ) + 1 ) → ( ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ( ( 𝑥 gcd 𝑦 ) = 1 ∧ 𝑧 = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ↔ ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ( ( 𝑥 gcd 𝑦 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) ) |
56 |
52 55 2
|
elab2 |
⊢ ( ( ( 𝑛 ↑ 2 ) + 1 ) ∈ 𝑌 ↔ ∃ 𝑥 ∈ ℤ ∃ 𝑦 ∈ ℤ ( ( 𝑥 gcd 𝑦 ) = 1 ∧ ( ( 𝑛 ↑ 2 ) + 1 ) = ( ( 𝑥 ↑ 2 ) + ( 𝑦 ↑ 2 ) ) ) ) |
57 |
51 56
|
sylibr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → ( ( 𝑛 ↑ 2 ) + 1 ) ∈ 𝑌 ) |
58 |
|
prmnn |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ ) |
59 |
58
|
ad2antrr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → 𝑃 ∈ ℕ ) |
60 |
|
simprr |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) |
61 |
31
|
zcnd |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → 𝑛 ∈ ℂ ) |
62 |
61
|
sqcld |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → ( 𝑛 ↑ 2 ) ∈ ℂ ) |
63 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
64 |
|
subneg |
⊢ ( ( ( 𝑛 ↑ 2 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑛 ↑ 2 ) − - 1 ) = ( ( 𝑛 ↑ 2 ) + 1 ) ) |
65 |
62 63 64
|
sylancl |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → ( ( 𝑛 ↑ 2 ) − - 1 ) = ( ( 𝑛 ↑ 2 ) + 1 ) ) |
66 |
60 65
|
breqtrd |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → 𝑃 ∥ ( ( 𝑛 ↑ 2 ) + 1 ) ) |
67 |
1 2
|
2sqlem10 |
⊢ ( ( ( ( 𝑛 ↑ 2 ) + 1 ) ∈ 𝑌 ∧ 𝑃 ∈ ℕ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) + 1 ) ) → 𝑃 ∈ 𝑆 ) |
68 |
57 59 66 67
|
syl3anc |
⊢ ( ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) ∧ ( 𝑛 ∈ ℤ ∧ 𝑃 ∥ ( ( 𝑛 ↑ 2 ) − - 1 ) ) ) → 𝑃 ∈ 𝑆 ) |
69 |
30 68
|
rexlimddv |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝑃 mod 4 ) = 1 ) → 𝑃 ∈ 𝑆 ) |