Metamath Proof Explorer


Theorem jm2.27dlem2

Description: Lemma for rmydioph . This theorem is used along with the next three to efficiently infer steps like 7 e. ( 1 ... ; 1 0 ) . (Contributed by Stefan O'Rear, 11-Oct-2014)

Ref Expression
Hypotheses jm2.27dlem2.1 𝐴 ∈ ( 1 ... 𝐵 )
jm2.27dlem2.2 𝐶 = ( 𝐵 + 1 )
jm2.27dlem2.3 𝐵 ∈ ℕ
Assertion jm2.27dlem2 𝐴 ∈ ( 1 ... 𝐶 )

Proof

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 ) → 𝐴𝐵 )
11 1 10 ax-mp 𝐴𝐵
12 letrp1 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵 ) → 𝐴 ≤ ( 𝐵 + 1 ) )
13 8 9 11 12 mp3an 𝐴 ≤ ( 𝐵 + 1 )
14 13 2 breqtrri 𝐴𝐶
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 ≤ 𝐴𝐴𝐶 ) ) )
21 15 19 20 mp2an ( 𝐴 ∈ ( 1 ... 𝐶 ) ↔ ( 𝐴 ∈ ℤ ∧ 1 ≤ 𝐴𝐴𝐶 ) )
22 5 7 14 21 mpbir3an 𝐴 ∈ ( 1 ... 𝐶 )