Theorem fzopred

Description: Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 . (Contributed by AV, 14-Jul-2020)

	Ref	Expression
Assertion	fzopred	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 ..^ 𝑁 ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fzolb	⊢ ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) )
2		fzofzp1	⊢ ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑀 + 1 ) ∈ ( 𝑀 ... 𝑁 ) )
3	1 2	sylbir	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 + 1 ) ∈ ( 𝑀 ... 𝑁 ) )
4		fzosplit	⊢ ( ( 𝑀 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝑀 ..^ 𝑁 ) = ( ( 𝑀 ..^ ( 𝑀 + 1 ) ) ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )
5	3 4	syl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 ..^ 𝑁 ) = ( ( 𝑀 ..^ ( 𝑀 + 1 ) ) ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )
6		fzosn	⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ..^ ( 𝑀 + 1 ) ) = { 𝑀 } )
7	6	3ad2ant1	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 ..^ ( 𝑀 + 1 ) ) = { 𝑀 } )
8	7	uneq1d	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( ( 𝑀 ..^ ( 𝑀 + 1 ) ) ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )
9	5 8	eqtrd	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 ..^ 𝑁 ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )