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 ( 𝜑𝐾𝑉 )
meetval2.x ( 𝜑𝑋𝐵 )
meetval2.y ( 𝜑𝑌𝐵 )
meetlem.e ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom )
Assertion meetlem ( 𝜑 → ( ( ( 𝑋 𝑌 ) 𝑋 ∧ ( 𝑋 𝑌 ) 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 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 ( 𝜑𝐾𝑉 )
5 meetval2.x ( 𝜑𝑋𝐵 )
6 meetval2.y ( 𝜑𝑌𝐵 )
7 meetlem.e ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom )
8 1 2 3 4 5 6 7 meeteu ( 𝜑 → ∃! 𝑥𝐵 ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) )
9 riotasbc ( ∃! 𝑥𝐵 ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) → [ ( 𝑥𝐵 ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) )
10 8 9 syl ( 𝜑[ ( 𝑥𝐵 ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) )
11 1 2 3 4 5 6 meetval2 ( 𝜑 → ( 𝑋 𝑌 ) = ( 𝑥𝐵 ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) ) )
12 11 sbceq1d ( 𝜑 → ( [ ( 𝑋 𝑌 ) / 𝑥 ] ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) ↔ [ ( 𝑥𝐵 ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) ) / 𝑥 ] ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) ) )
13 10 12 mpbird ( 𝜑[ ( 𝑋 𝑌 ) / 𝑥 ] ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 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 𝑋𝑥 𝑌 ) ↔ ( ( 𝑋 𝑌 ) 𝑋 ∧ ( 𝑋 𝑌 ) 𝑌 ) ) )
18 breq2 ( 𝑥 = ( 𝑋 𝑌 ) → ( 𝑧 𝑥𝑧 ( 𝑋 𝑌 ) ) )
19 18 imbi2d ( 𝑥 = ( 𝑋 𝑌 ) → ( ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ↔ ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 ( 𝑋 𝑌 ) ) ) )
20 19 ralbidv ( 𝑥 = ( 𝑋 𝑌 ) → ( ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ↔ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 ( 𝑋 𝑌 ) ) ) )
21 17 20 anbi12d ( 𝑥 = ( 𝑋 𝑌 ) → ( ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) ↔ ( ( ( 𝑋 𝑌 ) 𝑋 ∧ ( 𝑋 𝑌 ) 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 ( 𝑋 𝑌 ) ) ) ) )
22 14 21 sbcie ( [ ( 𝑋 𝑌 ) / 𝑥 ] ( ( 𝑥 𝑋𝑥 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 𝑥 ) ) ↔ ( ( ( 𝑋 𝑌 ) 𝑋 ∧ ( 𝑋 𝑌 ) 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 ( 𝑋 𝑌 ) ) ) )
23 13 22 sylib ( 𝜑 → ( ( ( 𝑋 𝑌 ) 𝑋 ∧ ( 𝑋 𝑌 ) 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 ( 𝑋 𝑌 ) ) ) )