Theorem xrletr

Description: Transitive law for ordering on extended reals. (Contributed by NM, 9-Feb-2006)

	Ref	Expression
Assertion	xrletr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 ≤ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		xrleloe	⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) ) )
2	1	3adant1	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) ) )
3	2	adantr	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) ) )
4		xrlelttr	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
5		xrltle	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < 𝐶 → 𝐴 ≤ 𝐶 ) )
6	5	3adant2	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < 𝐶 → 𝐴 ≤ 𝐶 ) )
7	4 6	syld	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 ≤ 𝐶 ) )
8	7	expdimp	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 < 𝐶 → 𝐴 ≤ 𝐶 ) )
9		breq2	⊢ ( 𝐵 = 𝐶 → ( 𝐴 ≤ 𝐵 ↔ 𝐴 ≤ 𝐶 ) )
10	9	biimpcd	⊢ ( 𝐴 ≤ 𝐵 → ( 𝐵 = 𝐶 → 𝐴 ≤ 𝐶 ) )
11	10	adantl	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 = 𝐶 → 𝐴 ≤ 𝐶 ) )
12	8 11	jaod	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) → 𝐴 ≤ 𝐶 ) )
13	3 12	sylbid	⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 ≤ 𝐶 → 𝐴 ≤ 𝐶 ) )
14	13	expimpd	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 ≤ 𝐶 ) )