Step |
Hyp |
Ref |
Expression |
1 |
|
znle2.y |
⊢ 𝑌 = ( ℤ/nℤ ‘ 𝑁 ) |
2 |
|
znle2.f |
⊢ 𝐹 = ( ( ℤRHom ‘ 𝑌 ) ↾ 𝑊 ) |
3 |
|
znle2.w |
⊢ 𝑊 = if ( 𝑁 = 0 , ℤ , ( 0 ..^ 𝑁 ) ) |
4 |
|
znle2.l |
⊢ ≤ = ( le ‘ 𝑌 ) |
5 |
|
znleval.x |
⊢ 𝑋 = ( Base ‘ 𝑌 ) |
6 |
1 2 3 4 5
|
znleval |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ) ) ) |
7 |
6
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ) ) ) |
8 |
|
3simpc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) |
9 |
8
|
biantrurd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ↔ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ) ) ) |
10 |
|
df-3an |
⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ) ↔ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ) ) |
11 |
9 10
|
bitr4di |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ↔ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ) ) ) |
12 |
7 11
|
bitr4d |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ≤ 𝐵 ↔ ( ◡ 𝐹 ‘ 𝐴 ) ≤ ( ◡ 𝐹 ‘ 𝐵 ) ) ) |