Step |
Hyp |
Ref |
Expression |
1 |
|
2lgslem2.n |
⊢ 𝑁 = ( ( ( 𝑃 − 1 ) / 2 ) − ( ⌊ ‘ ( 𝑃 / 4 ) ) ) |
2 |
|
nnnn0 |
⊢ ( 𝑃 ∈ ℕ → 𝑃 ∈ ℕ0 ) |
3 |
|
8nn |
⊢ 8 ∈ ℕ |
4 |
|
nnrp |
⊢ ( 8 ∈ ℕ → 8 ∈ ℝ+ ) |
5 |
3 4
|
ax-mp |
⊢ 8 ∈ ℝ+ |
6 |
|
modmuladdnn0 |
⊢ ( ( 𝑃 ∈ ℕ0 ∧ 8 ∈ ℝ+ ) → ( ( 𝑃 mod 8 ) = 7 → ∃ 𝑘 ∈ ℕ0 𝑃 = ( ( 𝑘 · 8 ) + 7 ) ) ) |
7 |
2 5 6
|
sylancl |
⊢ ( 𝑃 ∈ ℕ → ( ( 𝑃 mod 8 ) = 7 → ∃ 𝑘 ∈ ℕ0 𝑃 = ( ( 𝑘 · 8 ) + 7 ) ) ) |
8 |
|
simpr |
⊢ ( ( 𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
9 |
|
nn0cn |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ ) |
10 |
|
8cn |
⊢ 8 ∈ ℂ |
11 |
10
|
a1i |
⊢ ( 𝑘 ∈ ℕ0 → 8 ∈ ℂ ) |
12 |
9 11
|
mulcomd |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 · 8 ) = ( 8 · 𝑘 ) ) |
13 |
12
|
adantl |
⊢ ( ( 𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 · 8 ) = ( 8 · 𝑘 ) ) |
14 |
13
|
oveq1d |
⊢ ( ( 𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑘 · 8 ) + 7 ) = ( ( 8 · 𝑘 ) + 7 ) ) |
15 |
14
|
eqeq2d |
⊢ ( ( 𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) → ( 𝑃 = ( ( 𝑘 · 8 ) + 7 ) ↔ 𝑃 = ( ( 8 · 𝑘 ) + 7 ) ) ) |
16 |
15
|
biimpa |
⊢ ( ( ( 𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑃 = ( ( 𝑘 · 8 ) + 7 ) ) → 𝑃 = ( ( 8 · 𝑘 ) + 7 ) ) |
17 |
1
|
2lgslem3d |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝑃 = ( ( 8 · 𝑘 ) + 7 ) ) → 𝑁 = ( ( 2 · 𝑘 ) + 2 ) ) |
18 |
8 16 17
|
syl2an2r |
⊢ ( ( ( 𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑃 = ( ( 𝑘 · 8 ) + 7 ) ) → 𝑁 = ( ( 2 · 𝑘 ) + 2 ) ) |
19 |
|
oveq1 |
⊢ ( 𝑁 = ( ( 2 · 𝑘 ) + 2 ) → ( 𝑁 mod 2 ) = ( ( ( 2 · 𝑘 ) + 2 ) mod 2 ) ) |
20 |
|
2t1e2 |
⊢ ( 2 · 1 ) = 2 |
21 |
20
|
eqcomi |
⊢ 2 = ( 2 · 1 ) |
22 |
21
|
a1i |
⊢ ( 𝑘 ∈ ℕ0 → 2 = ( 2 · 1 ) ) |
23 |
22
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 2 · 𝑘 ) + 2 ) = ( ( 2 · 𝑘 ) + ( 2 · 1 ) ) ) |
24 |
|
2cnd |
⊢ ( 𝑘 ∈ ℕ0 → 2 ∈ ℂ ) |
25 |
|
1cnd |
⊢ ( 𝑘 ∈ ℕ0 → 1 ∈ ℂ ) |
26 |
|
adddi |
⊢ ( ( 2 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ ) → ( 2 · ( 𝑘 + 1 ) ) = ( ( 2 · 𝑘 ) + ( 2 · 1 ) ) ) |
27 |
26
|
eqcomd |
⊢ ( ( 2 ∈ ℂ ∧ 𝑘 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 2 · 𝑘 ) + ( 2 · 1 ) ) = ( 2 · ( 𝑘 + 1 ) ) ) |
28 |
24 9 25 27
|
syl3anc |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 2 · 𝑘 ) + ( 2 · 1 ) ) = ( 2 · ( 𝑘 + 1 ) ) ) |
29 |
9 25
|
addcld |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℂ ) |
30 |
24 29
|
mulcomd |
⊢ ( 𝑘 ∈ ℕ0 → ( 2 · ( 𝑘 + 1 ) ) = ( ( 𝑘 + 1 ) · 2 ) ) |
31 |
23 28 30
|
3eqtrd |
⊢ ( 𝑘 ∈ ℕ0 → ( ( 2 · 𝑘 ) + 2 ) = ( ( 𝑘 + 1 ) · 2 ) ) |
32 |
31
|
oveq1d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( 2 · 𝑘 ) + 2 ) mod 2 ) = ( ( ( 𝑘 + 1 ) · 2 ) mod 2 ) ) |
33 |
|
peano2nn0 |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℕ0 ) |
34 |
33
|
nn0zd |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℤ ) |
35 |
|
2rp |
⊢ 2 ∈ ℝ+ |
36 |
|
mulmod0 |
⊢ ( ( ( 𝑘 + 1 ) ∈ ℤ ∧ 2 ∈ ℝ+ ) → ( ( ( 𝑘 + 1 ) · 2 ) mod 2 ) = 0 ) |
37 |
34 35 36
|
sylancl |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( 𝑘 + 1 ) · 2 ) mod 2 ) = 0 ) |
38 |
32 37
|
eqtrd |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( 2 · 𝑘 ) + 2 ) mod 2 ) = 0 ) |
39 |
19 38
|
sylan9eqr |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝑁 = ( ( 2 · 𝑘 ) + 2 ) ) → ( 𝑁 mod 2 ) = 0 ) |
40 |
8 18 39
|
syl2an2r |
⊢ ( ( ( 𝑃 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑃 = ( ( 𝑘 · 8 ) + 7 ) ) → ( 𝑁 mod 2 ) = 0 ) |
41 |
40
|
rexlimdva2 |
⊢ ( 𝑃 ∈ ℕ → ( ∃ 𝑘 ∈ ℕ0 𝑃 = ( ( 𝑘 · 8 ) + 7 ) → ( 𝑁 mod 2 ) = 0 ) ) |
42 |
7 41
|
syld |
⊢ ( 𝑃 ∈ ℕ → ( ( 𝑃 mod 8 ) = 7 → ( 𝑁 mod 2 ) = 0 ) ) |
43 |
42
|
imp |
⊢ ( ( 𝑃 ∈ ℕ ∧ ( 𝑃 mod 8 ) = 7 ) → ( 𝑁 mod 2 ) = 0 ) |