Theorem ltletr

Description: Transitive law. (Contributed by NM, 25-Aug-1999)

	Ref	Expression
Assertion	ltletr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		leloe	⊢ ( ( 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		lttr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
4	3	expcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 < 𝐶 → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) ) )
5		breq2	⊢ ( 𝐵 = 𝐶 → ( 𝐴 < 𝐵 ↔ 𝐴 < 𝐶 ) )
6	5	biimpd	⊢ ( 𝐵 = 𝐶 → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) )
7	6	a1i	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 = 𝐶 → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) ) )
8	4 7	jaod	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) ) )
9	2 8	sylbid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 ≤ 𝐶 → ( 𝐴 < 𝐵 → 𝐴 < 𝐶 ) ) )
10	9	impcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) )