Metamath Proof Explorer


Theorem cdlemefs44

Description: Value of f_s(r) when r is an atom under pq and s is any atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefs45 instead TODO: FIX COMMENT. (Contributed by NM, 31-Mar-2013)

Ref Expression
Hypotheses cdlemef44.b 𝐵 = ( Base ‘ 𝐾 )
cdlemef44.l = ( le ‘ 𝐾 )
cdlemef44.j = ( join ‘ 𝐾 )
cdlemef44.m = ( meet ‘ 𝐾 )
cdlemef44.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemef44.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemef44.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdlemef44.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
cdlemef44.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) )
cdlemef44.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
cdlemefs44.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
cdlemefs44.i 𝐼 = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
Assertion cdlemefs44 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐹𝑅 ) = 𝑅 / 𝑠 𝑆 / 𝑡 𝐸 )

Proof

Step Hyp Ref Expression
1 cdlemef44.b 𝐵 = ( Base ‘ 𝐾 )
2 cdlemef44.l = ( le ‘ 𝐾 )
3 cdlemef44.j = ( join ‘ 𝐾 )
4 cdlemef44.m = ( meet ‘ 𝐾 )
5 cdlemef44.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemef44.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemef44.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdlemef44.d 𝐷 = ( ( 𝑡 𝑈 ) ( 𝑄 ( ( 𝑃 𝑡 ) 𝑊 ) ) )
9 cdlemef44.o 𝑂 = ( 𝑧𝐵𝑠𝐴 ( ( ¬ 𝑠 𝑊 ∧ ( 𝑠 ( 𝑥 𝑊 ) ) = 𝑥 ) → 𝑧 = ( if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝑠 / 𝑡 𝐷 ) ( 𝑥 𝑊 ) ) ) )
10 cdlemef44.f 𝐹 = ( 𝑥𝐵 ↦ if ( ( 𝑃𝑄 ∧ ¬ 𝑥 𝑊 ) , 𝑂 , 𝑥 ) )
11 cdlemefs44.e 𝐸 = ( ( 𝑃 𝑄 ) ( 𝐷 ( ( 𝑠 𝑡 ) 𝑊 ) ) )
12 cdlemefs44.i 𝐼 = ( 𝑦𝐵𝑡𝐴 ( ( ¬ 𝑡 𝑊 ∧ ¬ 𝑡 ( 𝑃 𝑄 ) ) → 𝑦 = 𝐸 ) )
13 eqid if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝑠 / 𝑡 𝐷 ) = if ( 𝑠 ( 𝑃 𝑄 ) , 𝐼 , 𝑠 / 𝑡 𝐷 )
14 eqid ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
15 eqid ( ( 𝑃 𝑄 ) ( ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) ( ( 𝑅 𝑆 ) 𝑊 ) ) ) = ( ( 𝑃 𝑄 ) ( ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) ( ( 𝑅 𝑆 ) 𝑊 ) ) )
16 1 2 3 4 5 6 7 8 11 12 13 9 10 14 15 cdlemefs31fv1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐹𝑅 ) = ( ( 𝑃 𝑄 ) ( ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) ( ( 𝑅 𝑆 ) 𝑊 ) ) ) )
17 simp22l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑅𝐴 )
18 simp23l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑆𝐴 )
19 8 11 14 15 cdleme31sde ( ( 𝑅𝐴𝑆𝐴 ) → 𝑅 / 𝑠 𝑆 / 𝑡 𝐸 = ( ( 𝑃 𝑄 ) ( ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) ( ( 𝑅 𝑆 ) 𝑊 ) ) ) )
20 17 18 19 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑅 / 𝑠 𝑆 / 𝑡 𝐸 = ( ( 𝑃 𝑄 ) ( ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) ( ( 𝑅 𝑆 ) 𝑊 ) ) ) )
21 16 20 eqtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐹𝑅 ) = 𝑅 / 𝑠 𝑆 / 𝑡 𝐸 )