Metamath Proof Explorer

Theorem cdlemg1idN

Description: Version of cdleme31id with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 18-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg1ltrn.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg1ltrn.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg1ltrn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg1ltrn.f ⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 )
cdlemg1ltrn.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg1id.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
Assertion cdlemg1idN ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 cdlemg1ltrn.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg1ltrn.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemg1ltrn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemg1ltrn.f ⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 )
5 cdlemg1ltrn.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 cdlemg1id.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
7 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
8 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
9 eqid ⊢ ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 )
10 eqid ⊢ ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
11 eqid ⊢ ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ( join ‘ 𝐾 ) ( ( 𝑠 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ( join ‘ 𝐾 ) ( ( 𝑠 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
12 eqid ⊢ ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ) → 𝑦 = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ( join ‘ 𝐾 ) ( ( 𝑠 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) , ⦋ 𝑠 / 𝑡 ⦌ ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) , 𝑥 ) ) = ( 𝑥 ∈ 𝐵 ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) , ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ) → 𝑦 = ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) ( ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ( join ‘ 𝐾 ) ( ( 𝑠 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) , ⦋ 𝑠 / 𝑡 ⦌ ( ( 𝑡 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ( meet ‘ 𝐾 ) ( 𝑄 ( join ‘ 𝐾 ) ( ( 𝑃 ( join ‘ 𝐾 ) 𝑡 ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) , 𝑥 ) )
13 6 1 7 8 2 3 9 10 11 12 5 4 cdlemg1idlemN ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑃 = 𝑄 ) → ( 𝐹 ‘ 𝑋 ) = 𝑋 )