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Metamath Proof Explorer

Theorem pltletr

Description: Transitive law for chained "less than" and "less than or equal to". ( psssstr analog.) (Contributed by NM, 2-Dec-2011)

Ref Expression
Hypotheses pltletr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
pltletr.l ⊢ ≤ = ( le ‘ 𝐾 )
pltletr.s ⊢ < = ( lt ‘ 𝐾 )
Assertion pltletr ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 < 𝑍 ) )

Proof

Step Hyp Ref Expression
1 pltletr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 pltletr.l ⊢ ≤ = ( le ‘ 𝐾 )
3 pltletr.s ⊢ < = ( lt ‘ 𝐾 )
4 1 2 3 pleval2 ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 ≤ 𝑍 ↔ ( 𝑌 < 𝑍 ∨ 𝑌 = 𝑍 ) ) )
5 4 3adant3r1 ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 ≤ 𝑍 ↔ ( 𝑌 < 𝑍 ∨ 𝑌 = 𝑍 ) ) )
6 5 adantr ⊢ ( ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 < 𝑌 ) → ( 𝑌 ≤ 𝑍 ↔ ( 𝑌 < 𝑍 ∨ 𝑌 = 𝑍 ) ) )
7 1 3 plttr ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) → 𝑋 < 𝑍 ) )
8 7 expdimp ⊢ ( ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 < 𝑌 ) → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) )
9 breq2 ⊢ ( 𝑌 = 𝑍 → ( 𝑋 < 𝑌 ↔ 𝑋 < 𝑍 ) )
10 9 biimpcd ⊢ ( 𝑋 < 𝑌 → ( 𝑌 = 𝑍 → 𝑋 < 𝑍 ) )
11 10 adantl ⊢ ( ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 < 𝑌 ) → ( 𝑌 = 𝑍 → 𝑋 < 𝑍 ) )
12 8 11 jaod ⊢ ( ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 < 𝑌 ) → ( ( 𝑌 < 𝑍 ∨ 𝑌 = 𝑍 ) → 𝑋 < 𝑍 ) )
13 6 12 sylbid ⊢ ( ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 < 𝑌 ) → ( 𝑌 ≤ 𝑍 → 𝑋 < 𝑍 ) )
14 13 expimpd ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 < 𝑍 ) )