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Metamath Proof Explorer

Theorem joinlem

Description: Lemma for join properties. (Contributed by NM, 16-Sep-2011) (Revised by NM, 12-Sep-2018)

Ref Expression
Hypotheses joinval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
joinval2.l ⊢ ≤ = ( le ‘ 𝐾 )
joinval2.j ⊢ ∨ = ( join ‘ 𝐾 )
joinval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
joinval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
joinval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
joinlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
Assertion joinlem ( 𝜑 → ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 joinval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 joinval2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 joinval2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 joinval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5 joinval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 joinval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 joinlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
8 1 2 3 4 5 6 7 joineu ⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) )
9 riotasbc ⊢ ( ∃! 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) → [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) / 𝑥 ] ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) )
10 8 9 syl ⊢ ( 𝜑 → [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) / 𝑥 ] ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) )
11 1 2 3 4 5 6 joinval2 ⊢ ( 𝜑 → ( 𝑋 ∨ 𝑌 ) = ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
12 11 sbceq1d ⊢ ( 𝜑 → ( [ ( 𝑋 ∨ 𝑌 ) / 𝑥 ] ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ↔ [ ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) / 𝑥 ] ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
13 10 12 mpbird ⊢ ( 𝜑 → [ ( 𝑋 ∨ 𝑌 ) / 𝑥 ] ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) )
14 ovex ⊢ ( 𝑋 ∨ 𝑌 ) ∈ V
15 breq2 ⊢ ( 𝑥 = ( 𝑋 ∨ 𝑌 ) → ( 𝑋 ≤ 𝑥 ↔ 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ) )
16 breq2 ⊢ ( 𝑥 = ( 𝑋 ∨ 𝑌 ) → ( 𝑌 ≤ 𝑥 ↔ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) )
17 15 16 anbi12d ⊢ ( 𝑥 = ( 𝑋 ∨ 𝑌 ) → ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ↔ ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ) )
18 breq1 ⊢ ( 𝑥 = ( 𝑋 ∨ 𝑌 ) → ( 𝑥 ≤ 𝑧 ↔ ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) )
19 18 imbi2d ⊢ ( 𝑥 = ( 𝑋 ∨ 𝑌 ) → ( ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ↔ ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) )
20 19 ralbidv ⊢ ( 𝑥 = ( 𝑋 ∨ 𝑌 ) → ( ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ↔ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) )
21 17 20 anbi12d ⊢ ( 𝑥 = ( 𝑋 ∨ 𝑌 ) → ( ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ↔ ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) ) )
22 14 21 sbcie ⊢ ( [ ( 𝑋 ∨ 𝑌 ) / 𝑥 ] ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ↔ ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) )
23 13 22 sylib ⊢ ( 𝜑 → ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) )