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Metamath Proof Explorer

Theorem meeteu

Description: Uniqueness of meet of elements in the domain. (Contributed by NM, 12-Sep-2018)

Ref Expression
Hypotheses meetval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
meetval2.l ⊢ ≤ = ( le ‘ 𝐾 )
meetval2.m ⊢ ∧ = ( meet ‘ 𝐾 )
meetval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
meetval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
meetval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
meetlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
Assertion meeteu ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 meetval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 meetval2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 meetval2.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 meetval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5 meetval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 meetval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 meetlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
8 eqid ⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 )
9 8 3 4 5 6 meetdef ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ ↔ { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) ) )
10 biid ⊢ ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) ↔ ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) )
11 4 adantr ⊢ ( ( 𝜑 ∧ { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) ) → 𝐾 ∈ 𝑉 )
12 simpr ⊢ ( ( 𝜑 ∧ { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) ) → { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) )
13 1 2 8 10 11 12 glbeu ⊢ ( ( 𝜑 ∧ { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) ) → ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) )
14 13 ex ⊢ ( 𝜑 → ( { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) → ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) ) )
15 1 2 3 4 5 6 meetval2lem ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) ↔ ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) )
16 5 6 15 syl2anc ⊢ ( 𝜑 → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) ↔ ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) )
17 16 reubidv ⊢ ( 𝜑 → ( ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑥 ≤ 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥 ) ) ↔ ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) )
18 14 17 sylibd ⊢ ( 𝜑 → ( { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) → ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) )
19 9 18 sylbid ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ → ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) )
20 7 19 mpd ⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) )