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 = ( 𝑓𝑇 ( 𝑓𝑃 ) = 𝑄 )
Assertion ltrniotacl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹𝑇 )

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 = ( 𝑓𝑇 ( 𝑓𝑃 ) = 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 ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) , ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑡𝐴 ( ( ¬ 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 ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧 ∈ ( Base ‘ 𝐾 ) ∀ 𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 ( join ‘ 𝐾 ) 𝑄 ) , ( 𝑦 ∈ ( Base ‘ 𝐾 ) ∀ 𝑡𝐴 ( ( ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → 𝐹𝑇 )