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Metamath Proof Explorer

Theorem letr

Description: Transitive law. (Contributed by NM, 12-Nov-1999)

Ref Expression
Assertion letr ( ( 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 2 adantr ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐵 < 𝐶 ∨ 𝐵 = 𝐶 ) ) )
4 lelttr ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 < 𝐶 ) → 𝐴 < 𝐶 ) )
5 ltle ⊢ ( ( 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 ) )