Step |
Hyp |
Ref |
Expression |
1 |
|
jm2.27dlem2.1 |
⊢ 𝐴 ∈ ( 1 ... 𝐵 ) |
2 |
|
jm2.27dlem2.2 |
⊢ 𝐶 = ( 𝐵 + 1 ) |
3 |
|
jm2.27dlem2.3 |
⊢ 𝐵 ∈ ℕ |
4 |
|
elfzelz |
⊢ ( 𝐴 ∈ ( 1 ... 𝐵 ) → 𝐴 ∈ ℤ ) |
5 |
1 4
|
ax-mp |
⊢ 𝐴 ∈ ℤ |
6 |
|
elfzle1 |
⊢ ( 𝐴 ∈ ( 1 ... 𝐵 ) → 1 ≤ 𝐴 ) |
7 |
1 6
|
ax-mp |
⊢ 1 ≤ 𝐴 |
8 |
5
|
zrei |
⊢ 𝐴 ∈ ℝ |
9 |
3
|
nnrei |
⊢ 𝐵 ∈ ℝ |
10 |
|
elfzle2 |
⊢ ( 𝐴 ∈ ( 1 ... 𝐵 ) → 𝐴 ≤ 𝐵 ) |
11 |
1 10
|
ax-mp |
⊢ 𝐴 ≤ 𝐵 |
12 |
|
letrp1 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ ( 𝐵 + 1 ) ) |
13 |
8 9 11 12
|
mp3an |
⊢ 𝐴 ≤ ( 𝐵 + 1 ) |
14 |
13 2
|
breqtrri |
⊢ 𝐴 ≤ 𝐶 |
15 |
|
1z |
⊢ 1 ∈ ℤ |
16 |
|
nnz |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℤ ) |
17 |
|
peano2z |
⊢ ( 𝐵 ∈ ℤ → ( 𝐵 + 1 ) ∈ ℤ ) |
18 |
3 16 17
|
mp2b |
⊢ ( 𝐵 + 1 ) ∈ ℤ |
19 |
2 18
|
eqeltri |
⊢ 𝐶 ∈ ℤ |
20 |
|
elfz1 |
⊢ ( ( 1 ∈ ℤ ∧ 𝐶 ∈ ℤ ) → ( 𝐴 ∈ ( 1 ... 𝐶 ) ↔ ( 𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶 ) ) ) |
21 |
15 19 20
|
mp2an |
⊢ ( 𝐴 ∈ ( 1 ... 𝐶 ) ↔ ( 𝐴 ∈ ℤ ∧ 1 ≤ 𝐴 ∧ 𝐴 ≤ 𝐶 ) ) |
22 |
5 7 14 21
|
mpbir3an |
⊢ 𝐴 ∈ ( 1 ... 𝐶 ) |