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 ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝐹𝑇 ) → ( 𝐹 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) )

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 ( ( 𝑝𝑞 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑝 𝑞 ) , ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑡𝐴 ( ( ¬ 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 ( ( 𝑝𝑞 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑝 𝑞 ) , ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑡𝐴 ( ( ¬ 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 ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝐹𝑇 ) → ( 𝐹 ‘ ( 𝑃 𝑄 ) ) = ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) )