Metamath Proof Explorer


Theorem cdlemefr45e

Description: Explicit expansion of cdlemefr45 . TODO: use to shorten cdlemefr45 uses? TODO: FIX COMMENT. (Contributed by NM, 10-Apr-2013)

Ref Expression
Hypotheses cdlemef45.b 𝐵 = ( Base ‘ 𝐾 )
cdlemef45.l = ( le ‘ 𝐾 )
cdlemef45.j = ( join ‘ 𝐾 )
cdlemef45.m = ( meet ‘ 𝐾 )
cdlemef45.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef45.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef45.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdlemef45.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdlemef45.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
Assertion cdlemefr45e ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝐹𝑅 ) = ( ( 𝑅 𝑈 ) ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemef45.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef45.l = ( le ‘ 𝐾 )
3 cdlemef45.j = ( join ‘ 𝐾 )
4 cdlemef45.m = ( meet ‘ 𝐾 )
5 cdlemef45.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef45.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef45.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdlemef45.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
9 cdlemef45.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) ) , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) ) , 𝑥 ) )
10 1 2 3 4 5 6 7 8 9 cdlemefr45 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝐹𝑅 ) = 𝑅 / 𝑡 𝐷 )
11 simp2rl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅𝐴 )
12 eqid ( ( 𝑅 𝑈 ) ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ) = ( ( 𝑅 𝑈 ) ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) )
13 8 12 cdleme31sc ( 𝑅𝐴 𝑅 / 𝑡 𝐷 = ( ( 𝑅 𝑈 ) ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ) )
14 11 13 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅 / 𝑡 𝐷 = ( ( 𝑅 𝑈 ) ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ) )
15 10 14 eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝐹𝑅 ) = ( ( 𝑅 𝑈 ) ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ) )