Step |
Hyp |
Ref |
Expression |
1 |
|
uzind3.1 |
⊢ ( 𝑗 = 𝑀 → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
uzind3.2 |
⊢ ( 𝑗 = 𝑚 → ( 𝜑 ↔ 𝜒 ) ) |
3 |
|
uzind3.3 |
⊢ ( 𝑗 = ( 𝑚 + 1 ) → ( 𝜑 ↔ 𝜃 ) ) |
4 |
|
uzind3.4 |
⊢ ( 𝑗 = 𝑁 → ( 𝜑 ↔ 𝜏 ) ) |
5 |
|
uzind3.5 |
⊢ ( 𝑀 ∈ ℤ → 𝜓 ) |
6 |
|
uzind3.6 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑚 ∈ { 𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘 } ) → ( 𝜒 → 𝜃 ) ) |
7 |
|
breq2 |
⊢ ( 𝑘 = 𝑁 → ( 𝑀 ≤ 𝑘 ↔ 𝑀 ≤ 𝑁 ) ) |
8 |
7
|
elrab |
⊢ ( 𝑁 ∈ { 𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘 } ↔ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) |
9 |
|
breq2 |
⊢ ( 𝑘 = 𝑚 → ( 𝑀 ≤ 𝑘 ↔ 𝑀 ≤ 𝑚 ) ) |
10 |
9
|
elrab |
⊢ ( 𝑚 ∈ { 𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘 } ↔ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ) ) |
11 |
10 6
|
sylan2br |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ) ) → ( 𝜒 → 𝜃 ) ) |
12 |
11
|
3impb |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑀 ≤ 𝑚 ) → ( 𝜒 → 𝜃 ) ) |
13 |
1 2 3 4 5 12
|
uzind |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) → 𝜏 ) |
14 |
13
|
3expb |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁 ) ) → 𝜏 ) |
15 |
8 14
|
sylan2b |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ { 𝑘 ∈ ℤ ∣ 𝑀 ≤ 𝑘 } ) → 𝜏 ) |