ioounsn

Description: The union of an open interval with its upper endpoint is a left-open right-closed interval. (Contributed by Jon Pennant, 8-Jun-2019)

		Ref	Expression
	Assertion	ioounsn	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐵 } ) = ( 𝐴 (,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simp2	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ^* )
2		iccid	⊢ ( 𝐵 ∈ ℝ^* → ( 𝐵 [,] 𝐵 ) = { 𝐵 } )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → ( 𝐵 [,] 𝐵 ) = { 𝐵 } )
4	3	uneq2d	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐵 } ) )
5		simp1	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ^* )
6		simp3	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 )
7	1	xrleidd	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → 𝐵 ≤ 𝐵 )
8		df-ioo	⊢ (,) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑧 ∈ ℝ^* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
9		df-icc	⊢ [,] = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑧 ∈ ℝ^* ∣ ( 𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
10		xrlenlt	⊢ ( ( 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^* ) → ( 𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵 ) )
11		df-ioc	⊢ (,] = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑧 ∈ ℝ^* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦 ) } )
12		simpl1	⊢ ( ( ( 𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → 𝑤 ∈ ℝ^* )
13		simpl2	⊢ ( ( ( 𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → 𝐵 ∈ ℝ^* )
14		simprl	⊢ ( ( ( 𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → 𝑤 < 𝐵 )
15	12 13 14	xrltled	⊢ ( ( ( 𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → 𝑤 ≤ 𝐵 )
16	15	ex	⊢ ( ( 𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) → ( ( 𝑤 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) → 𝑤 ≤ 𝐵 ) )
17		xrltletr	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^* ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝑤 ) → 𝐴 < 𝑤 ) )
18	8 9 10 11 16 17	ixxun	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵 ) ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( 𝐴 (,] 𝐵 ) )
19	5 1 1 6 7 18	syl32anc	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ ( 𝐵 [,] 𝐵 ) ) = ( 𝐴 (,] 𝐵 ) )
20	4 19	eqtr3d	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐵 } ) = ( 𝐴 (,] 𝐵 ) )

Metamath Proof Explorer

Theorem ioounsn

Proof