Proof of Theorem zltp1le
| Step | Hyp | Ref
| Expression |
| 1 | | nnge1 12294 |
. . . 4
⊢ ((𝑁 − 𝑀) ∈ ℕ → 1 ≤ (𝑁 − 𝑀)) |
| 2 | 1 | a1i 11 |
. . 3
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 − 𝑀) ∈ ℕ → 1 ≤ (𝑁 − 𝑀))) |
| 3 | | znnsub 12663 |
. . 3
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ)) |
| 4 | | zre 12617 |
. . . 4
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
ℝ) |
| 5 | | zre 12617 |
. . . 4
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
ℝ) |
| 6 | | 1re 11261 |
. . . . 5
⊢ 1 ∈
ℝ |
| 7 | | leaddsub2 11740 |
. . . . 5
⊢ ((𝑀 ∈ ℝ ∧ 1 ∈
ℝ ∧ 𝑁 ∈
ℝ) → ((𝑀 + 1)
≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) |
| 8 | 6, 7 | mp3an2 1451 |
. . . 4
⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 1) ≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) |
| 9 | 4, 5, 8 | syl2an 596 |
. . 3
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 1) ≤ 𝑁 ↔ 1 ≤ (𝑁 − 𝑀))) |
| 10 | 2, 3, 9 | 3imtr4d 294 |
. 2
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 → (𝑀 + 1) ≤ 𝑁)) |
| 11 | 4 | adantr 480 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈
ℝ) |
| 12 | 11 | ltp1d 12198 |
. . 3
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 < (𝑀 + 1)) |
| 13 | | peano2re 11434 |
. . . . 5
⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈
ℝ) |
| 14 | 11, 13 | syl 17 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 1) ∈
ℝ) |
| 15 | 5 | adantl 481 |
. . . 4
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈
ℝ) |
| 16 | | ltletr 11353 |
. . . 4
⊢ ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 < (𝑀 + 1) ∧ (𝑀 + 1) ≤ 𝑁) → 𝑀 < 𝑁)) |
| 17 | 11, 14, 15, 16 | syl3anc 1373 |
. . 3
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 < (𝑀 + 1) ∧ (𝑀 + 1) ≤ 𝑁) → 𝑀 < 𝑁)) |
| 18 | 12, 17 | mpand 695 |
. 2
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 1) ≤ 𝑁 → 𝑀 < 𝑁)) |
| 19 | 10, 18 | impbid 212 |
1
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |