Theorem fzosplit

Description: Split a half-open integer range in half. (Contributed by Stefan O'Rear, 14-Aug-2015)

	Ref	Expression
Assertion	fzosplit	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → ( 𝐵 ..^ 𝐶 ) = ( ( 𝐵 ..^ 𝐷 ) ∪ ( 𝐷 ..^ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpr	⊢ ( ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 ..^ 𝐶 ) ) → 𝑥 ∈ ( 𝐵 ..^ 𝐶 ) )
2		elfzelz	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → 𝐷 ∈ ℤ )
3	2	adantr	⊢ ( ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 ..^ 𝐶 ) ) → 𝐷 ∈ ℤ )
4		fzospliti	⊢ ( ( 𝑥 ∈ ( 𝐵 ..^ 𝐶 ) ∧ 𝐷 ∈ ℤ ) → ( 𝑥 ∈ ( 𝐵 ..^ 𝐷 ) ∨ 𝑥 ∈ ( 𝐷 ..^ 𝐶 ) ) )
5	1 3 4	syl2anc	⊢ ( ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 ..^ 𝐶 ) ) → ( 𝑥 ∈ ( 𝐵 ..^ 𝐷 ) ∨ 𝑥 ∈ ( 𝐷 ..^ 𝐶 ) ) )
6		elun	⊢ ( 𝑥 ∈ ( ( 𝐵 ..^ 𝐷 ) ∪ ( 𝐷 ..^ 𝐶 ) ) ↔ ( 𝑥 ∈ ( 𝐵 ..^ 𝐷 ) ∨ 𝑥 ∈ ( 𝐷 ..^ 𝐶 ) ) )
7	5 6	sylibr	⊢ ( ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 ..^ 𝐶 ) ) → 𝑥 ∈ ( ( 𝐵 ..^ 𝐷 ) ∪ ( 𝐷 ..^ 𝐶 ) ) )
8	7	ex	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → ( 𝑥 ∈ ( 𝐵 ..^ 𝐶 ) → 𝑥 ∈ ( ( 𝐵 ..^ 𝐷 ) ∪ ( 𝐷 ..^ 𝐶 ) ) ) )
9	8	ssrdv	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → ( 𝐵 ..^ 𝐶 ) ⊆ ( ( 𝐵 ..^ 𝐷 ) ∪ ( 𝐷 ..^ 𝐶 ) ) )
10		elfzuz3	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → 𝐶 ∈ ( ℤ≥ ‘ 𝐷 ) )
11		fzoss2	⊢ ( 𝐶 ∈ ( ℤ≥ ‘ 𝐷 ) → ( 𝐵 ..^ 𝐷 ) ⊆ ( 𝐵 ..^ 𝐶 ) )
12	10 11	syl	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → ( 𝐵 ..^ 𝐷 ) ⊆ ( 𝐵 ..^ 𝐶 ) )
13		elfzuz	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → 𝐷 ∈ ( ℤ≥ ‘ 𝐵 ) )
14		fzoss1	⊢ ( 𝐷 ∈ ( ℤ≥ ‘ 𝐵 ) → ( 𝐷 ..^ 𝐶 ) ⊆ ( 𝐵 ..^ 𝐶 ) )
15	13 14	syl	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → ( 𝐷 ..^ 𝐶 ) ⊆ ( 𝐵 ..^ 𝐶 ) )
16	12 15	unssd	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → ( ( 𝐵 ..^ 𝐷 ) ∪ ( 𝐷 ..^ 𝐶 ) ) ⊆ ( 𝐵 ..^ 𝐶 ) )
17	9 16	eqssd	⊢ ( 𝐷 ∈ ( 𝐵 ... 𝐶 ) → ( 𝐵 ..^ 𝐶 ) = ( ( 𝐵 ..^ 𝐷 ) ∪ ( 𝐷 ..^ 𝐶 ) ) )