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Metamath Proof Explorer

Theorem cdlemg2jOLDN

Description: TODO: Replace this with ltrnj . (Contributed by NM, 22-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg2inv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg2inv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg2j.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg2j.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg2j.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cdlemg2jOLDN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg2inv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 cdlemg2inv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 cdlemg2j.l ⊢ ≤ = ( le ‘ 𝐾 )
4 cdlemg2j.j ⊢ ∨ = ( join ‘ 𝐾 )
5 cdlemg2j.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
8 eqid ⊢ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 )
9 eqid ⊢ ( ( 𝑡 ∨ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝑡 ∨ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
10 eqid ⊢ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ∨ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( 𝑠 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ∨ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( 𝑠 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
11 eqid ⊢ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ∨ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( 𝑠 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) , ⦋ 𝑠 / 𝑡 ⦌ ( ( 𝑡 ∨ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ∨ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) , 𝑥 ) ) = ( 𝑥 ∈ ( Base ‘ 𝐾 ) ↦ if ( ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑝 ∨ 𝑞 ) , ( ℩ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑝 ∨ 𝑞 ) ) → 𝑦 = ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ∨ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ∨ ( ( 𝑠 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) , ⦋ 𝑠 / 𝑡 ⦌ ( ( 𝑡 ∨ ( ( 𝑝 ∨ 𝑞 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑞 ∨ ( ( 𝑝 ∨ 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ∨ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) , 𝑥 ) )
12 6 3 4 7 5 1 2 8 9 10 11 cdlemg2jlemOLDN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )