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Metamath Proof Explorer

Theorem meetlem

Description: Lemma for meet properties. (Contributed by NM, 16-Sep-2011) (Revised 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 meetlem ( 𝜑 → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) )

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 1 2 3 4 5 6 7 meeteu ⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) )
9 riotasbc ⊢ ( ∃! 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) → [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) )
10 8 9 syl ⊢ ( 𝜑 → [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) )
11 1 2 3 4 5 6 meetval2 ⊢ ( 𝜑 → ( 𝑋 ∧ 𝑌 ) = ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) )
12 11 sbceq1d ⊢ ( 𝜑 → ( [ ( 𝑋 ∧ 𝑌 ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ↔ [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ) )
13 10 12 mpbird ⊢ ( 𝜑 → [ ( 𝑋 ∧ 𝑌 ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) )
14 ovex ⊢ ( 𝑋 ∧ 𝑌 ) ∈ V
15 breq1 ⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( 𝑥 ≤ 𝑋 ↔ ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ) )
16 breq1 ⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( 𝑥 ≤ 𝑌 ↔ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) )
17 15 16 anbi12d ⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ↔ ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ) )
18 breq2 ⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( 𝑧 ≤ 𝑥 ↔ 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) )
19 18 imbi2d ⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ↔ ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) )
20 19 ralbidv ⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ↔ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) )
21 17 20 anbi12d ⊢ ( 𝑥 = ( 𝑋 ∧ 𝑌 ) → ( ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ↔ ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) ) )
22 14 21 sbcie ⊢ ( [ ( 𝑋 ∧ 𝑌 ) / 𝑥 ] ( ( 𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ 𝑥 ) ) ↔ ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) )
23 13 22 sylib ⊢ ( 𝜑 → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) )