Theorem xrletrd

Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015)

	Ref	Expression
Hypotheses	xrlttrd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xrlttrd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	xrlttrd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
	xrletrd.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
	xrletrd.5	⊢ ( 𝜑 → 𝐵 ≤ 𝐶 )
Assertion	xrletrd	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		xrlttrd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xrlttrd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		xrlttrd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ* )
4		xrletrd.4	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
5		xrletrd.5	⊢ ( 𝜑 → 𝐵 ≤ 𝐶 )
6		xrletr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 ≤ 𝐶 ) )
7	1 2 3 6	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 ≤ 𝐶 ) )
8	4 5 7	mp2and	⊢ ( 𝜑 → 𝐴 ≤ 𝐶 )