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Metamath Proof Explorer

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 )