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 ) )