Theorem plelttr

Description: Transitive law for chained "less than or equal to" and "less than". ( sspsstr analog.) (Contributed by NM, 2-May-2012)

	Ref	Expression
Hypotheses	pltletr.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pltletr.l	⊢ ≤ = ( le ‘ 𝐾 )
	pltletr.s	⊢ < = ( lt ‘ 𝐾 )
Assertion	plelttr	⊢ ( ( 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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) ) )
5	4	3adant3r3	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) ) )
6	1 3	plttr	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) → 𝑋 < 𝑍 ) )
7	6	expd	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 < 𝑌 → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) )
8		breq1	⊢ ( 𝑋 = 𝑌 → ( 𝑋 < 𝑍 ↔ 𝑌 < 𝑍 ) )
9	8	biimprd	⊢ ( 𝑋 = 𝑌 → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) )
10	9	a1i	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 = 𝑌 → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) )
11	7 10	jaod	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) )
12	5 11	sylbid	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 → ( 𝑌 < 𝑍 → 𝑋 < 𝑍 ) ) )
13	12	impd	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 < 𝑍 ) → 𝑋 < 𝑍 ) )