Step |
Hyp |
Ref |
Expression |
1 |
|
gausslemma2dlem0.p |
⊢ ( 𝜑 → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) |
2 |
|
gausslemma2dlem0.m |
⊢ 𝑀 = ( ⌊ ‘ ( 𝑃 / 4 ) ) |
3 |
|
gausslemma2dlem0.h |
⊢ 𝐻 = ( ( 𝑃 − 1 ) / 2 ) |
4 |
|
eldifsn |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) ↔ ( 𝑃 ∈ ℙ ∧ 𝑃 ≠ 2 ) ) |
5 |
|
prm23ge5 |
⊢ ( 𝑃 ∈ ℙ → ( 𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) |
6 |
|
eqneqall |
⊢ ( 𝑃 = 2 → ( 𝑃 ≠ 2 → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) ) |
7 |
|
orc |
⊢ ( 𝑃 = 3 → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) |
8 |
7
|
a1d |
⊢ ( 𝑃 = 3 → ( 𝑃 ≠ 2 → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) ) |
9 |
|
olc |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 5 ) → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) |
10 |
9
|
a1d |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 5 ) → ( 𝑃 ≠ 2 → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) ) |
11 |
6 8 10
|
3jaoi |
⊢ ( ( 𝑃 = 2 ∨ 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) → ( 𝑃 ≠ 2 → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) ) |
12 |
5 11
|
syl |
⊢ ( 𝑃 ∈ ℙ → ( 𝑃 ≠ 2 → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) ) |
13 |
12
|
imp |
⊢ ( ( 𝑃 ∈ ℙ ∧ 𝑃 ≠ 2 ) → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) |
14 |
4 13
|
sylbi |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) ) |
15 |
|
fldiv4p1lem1div2 |
⊢ ( ( 𝑃 = 3 ∨ 𝑃 ∈ ( ℤ≥ ‘ 5 ) ) → ( ( ⌊ ‘ ( 𝑃 / 4 ) ) + 1 ) ≤ ( ( 𝑃 − 1 ) / 2 ) ) |
16 |
1 14 15
|
3syl |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 𝑃 / 4 ) ) + 1 ) ≤ ( ( 𝑃 − 1 ) / 2 ) ) |
17 |
2
|
oveq1i |
⊢ ( 𝑀 + 1 ) = ( ( ⌊ ‘ ( 𝑃 / 4 ) ) + 1 ) |
18 |
16 17 3
|
3brtr4g |
⊢ ( 𝜑 → ( 𝑀 + 1 ) ≤ 𝐻 ) |