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Metamath Proof Explorer

Theorem mrelatlubALT

Description: Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015) (Proof shortened by Zhi Wang, 29-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses mreclatGOOD.i ⊢ 𝐼 = ( toInc ‘ 𝐶 )
mrelatlubALT.f ⊢ 𝐹 = ( mrCls ‘ 𝐶 )
mrelatlubALT.l ⊢ 𝐿 = ( lub ‘ 𝐼 )
Assertion mrelatlubALT ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ( 𝐿 ‘ 𝑈 ) = ( 𝐹 ‘ ∪ 𝑈 ) )

Proof

Step Hyp Ref Expression
1 mreclatGOOD.i ⊢ 𝐼 = ( toInc ‘ 𝐶 )
2 mrelatlubALT.f ⊢ 𝐹 = ( mrCls ‘ 𝐶 )
3 mrelatlubALT.l ⊢ 𝐿 = ( lub ‘ 𝐼 )
4 simpl ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → 𝐶 ∈ ( Moore ‘ 𝑋 ) )
5 simpr ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → 𝑈 ⊆ 𝐶 )
6 3 a1i ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → 𝐿 = ( lub ‘ 𝐼 ) )
7 mreuniss ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ∪ 𝑈 ⊆ 𝑋 )
8 2 mrcval ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∪ 𝑈 ⊆ 𝑋 ) → ( 𝐹 ‘ ∪ 𝑈 ) = ∩ { 𝑥 ∈ 𝐶 ∣ ∪ 𝑈 ⊆ 𝑥 } )
9 7 8 syldan ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ( 𝐹 ‘ ∪ 𝑈 ) = ∩ { 𝑥 ∈ 𝐶 ∣ ∪ 𝑈 ⊆ 𝑥 } )
10 2 mrccl ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∪ 𝑈 ⊆ 𝑋 ) → ( 𝐹 ‘ ∪ 𝑈 ) ∈ 𝐶 )
11 7 10 syldan ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ( 𝐹 ‘ ∪ 𝑈 ) ∈ 𝐶 )
12 1 4 5 6 9 11 ipolub ⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ( 𝐿 ‘ 𝑈 ) = ( 𝐹 ‘ ∪ 𝑈 ) )