Metamath Proof Explorer

Theorem ltrniotacl

Description: Version of cdleme50ltrn with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 17-Apr-2013)

Ref Expression
Hypotheses ltrniotaval.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrniotaval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrniotaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrniotaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
ltrniotaval.f ⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 )
Assertion ltrniotacl ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )

Proof

Step Hyp Ref Expression
1 ltrniotaval.l ⊢ ≤ = ( le ‘ 𝐾 )
2 ltrniotaval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 ltrniotaval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 ltrniotaval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 ltrniotaval.f ⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑄 )
6 eqid ⊢ ( Base ‘ 𝐾 ) = ( 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 ∈ ( Base ‘ 𝐾 ) ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) , ( ℩ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 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 ∈ ( Base ‘ 𝐾 ) ↦ if ( ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊 ) , ( ℩ 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ≤ ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) , ( ℩ 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 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 4 5 cdlemg1ltrnlem ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )