iocinif

Description: Relate intersection of two open-below, closed-above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017)

		Ref	Expression
	Assertion	iocinif	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = if ( 𝐴 < 𝐵 , ( 𝐵 (,] 𝐶 ) , ( 𝐴 (,] 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		exmid	⊢ ( 𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵 )
2		xrltle	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) → ( 𝐴 < 𝐵 → 𝐴 ≤ 𝐵 ) )
3	2	imp	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ) ∧ 𝐴 < 𝐵 ) → 𝐴 ≤ 𝐵 )
4	3	3adantl3	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ 𝐴 < 𝐵 ) → 𝐴 ≤ 𝐵 )
5		iocinioc2	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) )
6	4 5	syldan	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ 𝐴 < 𝐵 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) )
7	6	ex	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) → ( 𝐴 < 𝐵 → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) )
8	7	ancld	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) → ( 𝐴 < 𝐵 → ( 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ) )
9		simpl2	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ^* )
10		simpl1	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ^* )
11		simpr	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ ¬ 𝐴 < 𝐵 ) → ¬ 𝐴 < 𝐵 )
12		xrlenlt	⊢ ( ( 𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ) → ( 𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵 ) )
13	12	biimpar	⊢ ( ( ( 𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐵 ≤ 𝐴 )
14	9 10 11 13	syl21anc	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ ¬ 𝐴 < 𝐵 ) → 𝐵 ≤ 𝐴 )
15		3ancoma	⊢ ( ( 𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ↔ ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) )
16		incom	⊢ ( ( 𝐵 (,] 𝐶 ) ∩ ( 𝐴 (,] 𝐶 ) ) = ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) )
17		iocinioc2	⊢ ( ( ( 𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ 𝐵 ≤ 𝐴 ) → ( ( 𝐵 (,] 𝐶 ) ∩ ( 𝐴 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) )
18	16 17	eqtr3id	⊢ ( ( ( 𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ 𝐵 ≤ 𝐴 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) )
19	15 18	sylanbr	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ 𝐵 ≤ 𝐴 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) )
20	14 19	syldan	⊢ ( ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) ∧ ¬ 𝐴 < 𝐵 ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) )
21	20	ex	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) → ( ¬ 𝐴 < 𝐵 → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) )
22	21	ancld	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) → ( ¬ 𝐴 < 𝐵 → ( ¬ 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) )
23	8 22	orim12d	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) → ( ( 𝐴 < 𝐵 ∨ ¬ 𝐴 < 𝐵 ) → ( ( 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ∨ ( ¬ 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) ) )
24	1 23	mpi	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) → ( ( 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ∨ ( ¬ 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) )
25		eqif	⊢ ( ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = if ( 𝐴 < 𝐵 , ( 𝐵 (,] 𝐶 ) , ( 𝐴 (,] 𝐶 ) ) ↔ ( ( 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐵 (,] 𝐶 ) ) ∨ ( ¬ 𝐴 < 𝐵 ∧ ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = ( 𝐴 (,] 𝐶 ) ) ) )
26	24 25	sylibr	⊢ ( ( 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ) → ( ( 𝐴 (,] 𝐶 ) ∩ ( 𝐵 (,] 𝐶 ) ) = if ( 𝐴 < 𝐵 , ( 𝐵 (,] 𝐶 ) , ( 𝐴 (,] 𝐶 ) ) )

Metamath Proof Explorer

Theorem iocinif

Proof