Theorem eluzuzle

Description: An integer in an upper set of integers is an element of an upper set of integers with a smaller bound. (Contributed by Alexander van der Vekens, 17-Jun-2018)

	Ref	Expression
Assertion	eluzuzle	⊢ ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) → ( 𝐶 ∈ ( ℤ≥ ‘ 𝐴 ) → 𝐶 ∈ ( ℤ≥ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eluz2	⊢ ( 𝐶 ∈ ( ℤ≥ ‘ 𝐴 ) ↔ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) )
2		simpll	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐵 ∈ ℤ )
3		simpr2	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐶 ∈ ℤ )
4		zre	⊢ ( 𝐵 ∈ ℤ → 𝐵 ∈ ℝ )
5	4	ad2antrr	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐵 ∈ ℝ )
6		zre	⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℝ )
7	6	3ad2ant1	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) → 𝐴 ∈ ℝ )
8	7	adantl	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐴 ∈ ℝ )
9		zre	⊢ ( 𝐶 ∈ ℤ → 𝐶 ∈ ℝ )
10	9	3ad2ant2	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) → 𝐶 ∈ ℝ )
11	10	adantl	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐶 ∈ ℝ )
12		simplr	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐵 ≤ 𝐴 )
13		simpr3	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐴 ≤ 𝐶 )
14	5 8 11 12 13	letrd	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐵 ≤ 𝐶 )
15		eluz2	⊢ ( 𝐶 ∈ ( ℤ≥ ‘ 𝐵 ) ↔ ( 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ≤ 𝐶 ) )
16	2 3 14 15	syl3anbrc	⊢ ( ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) ∧ ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) ) → 𝐶 ∈ ( ℤ≥ ‘ 𝐵 ) )
17	16	ex	⊢ ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) → ( ( 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐴 ≤ 𝐶 ) → 𝐶 ∈ ( ℤ≥ ‘ 𝐵 ) ) )
18	1 17	syl5bi	⊢ ( ( 𝐵 ∈ ℤ ∧ 𝐵 ≤ 𝐴 ) → ( 𝐶 ∈ ( ℤ≥ ‘ 𝐴 ) → 𝐶 ∈ ( ℤ≥ ‘ 𝐵 ) ) )