Theorem lemaxle

Description: A real number which is less than or equal to a second real number is less than or equal to the maximum/supremum of the second real number and a third real number. (Contributed by AV, 8-Jun-2021)

	Ref	Expression
Assertion	lemaxle	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		max2	⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐵 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) )
2	1	ancoms	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → 𝐵 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) )
3	2	adantr	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ∈ ℝ ) → 𝐵 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) )
4		simpr	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ∈ ℝ ) → 𝐴 ∈ ℝ )
5		simpll	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ∈ ℝ ) → 𝐵 ∈ ℝ )
6		ifcl	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) ∈ ℝ )
7	6	adantr	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ∈ ℝ ) → if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) ∈ ℝ )
8		letr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) ) → 𝐴 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) ) )
9	4 5 7 8	syl3anc	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) ) → 𝐴 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) ) )
10	3 9	mpan2d	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ∈ ℝ ) → ( 𝐴 ≤ 𝐵 → 𝐴 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) ) )
11	10	3impia	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ if ( 𝐶 ≤ 𝐵 , 𝐵 , 𝐶 ) )