Theorem op01dm

Description: Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018)

	Ref	Expression
Hypotheses	op01dm.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	op01dm.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
	op01dm.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
Assertion	op01dm	⊢ ( 𝐾 ∈ OP → ( 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		op01dm.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		op01dm.u	⊢ 𝑈 = ( lub ‘ 𝐾 )
3		op01dm.g	⊢ 𝐺 = ( glb ‘ 𝐾 )
4		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5		eqid	⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
6		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
7		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8		eqid	⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
9		eqid	⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
10	1 2 3 4 5 6 7 8 9	isopos	⊢ ( 𝐾 ∈ OP ↔ ( ( 𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ∈ 𝐵 ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = 𝑥 ∧ ( 𝑥 ( le ‘ 𝐾 ) 𝑦 → ( ( oc ‘ 𝐾 ) ‘ 𝑦 ) ( le ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) ) ∧ ( 𝑥 ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = ( 1. ‘ 𝐾 ) ∧ ( 𝑥 ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = ( 0. ‘ 𝐾 ) ) ) )
11		simpl	⊢ ( ( ( 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ∈ 𝐵 ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = 𝑥 ∧ ( 𝑥 ( le ‘ 𝐾 ) 𝑦 → ( ( oc ‘ 𝐾 ) ‘ 𝑦 ) ( le ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) ) ∧ ( 𝑥 ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = ( 1. ‘ 𝐾 ) ∧ ( 𝑥 ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = ( 0. ‘ 𝐾 ) ) ) → ( 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺 ) )
12	11	3adantl1	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺 ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ∈ 𝐵 ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = 𝑥 ∧ ( 𝑥 ( le ‘ 𝐾 ) 𝑦 → ( ( oc ‘ 𝐾 ) ‘ 𝑦 ) ( le ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) ) ∧ ( 𝑥 ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = ( 1. ‘ 𝐾 ) ∧ ( 𝑥 ( meet ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑥 ) ) = ( 0. ‘ 𝐾 ) ) ) → ( 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺 ) )
13	10 12	sylbi	⊢ ( 𝐾 ∈ OP → ( 𝐵 ∈ dom 𝑈 ∧ 𝐵 ∈ dom 𝐺 ) )