Metamath Proof Explorer


Theorem cdleme43bN

Description: Lemma for Lemma E in Crawley p. 113. g(s) is an atom not under w. (Contributed by NM, 20-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme43.b 𝐵 = ( Base ‘ 𝐾 )
cdleme43.l = ( le ‘ 𝐾 )
cdleme43.j = ( join ‘ 𝐾 )
cdleme43.m = ( meet ‘ 𝐾 )
cdleme43.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme43.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme43.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme43.x 𝑋 = ( ( 𝑄 𝑃 ) 𝑊 )
cdleme43.c 𝐶 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme43.f 𝑍 = ( ( 𝑃 𝑄 ) ( 𝐶 ( ( 𝑅 𝑆 ) 𝑊 ) ) )
cdleme43.d 𝐷 = ( ( 𝑆 𝑋 ) ( 𝑃 ( ( 𝑄 𝑆 ) 𝑊 ) ) )
cdleme43.g 𝐺 = ( ( 𝑄 𝑃 ) ( 𝐷 ( ( 𝑍 𝑆 ) 𝑊 ) ) )
cdleme43.e 𝐸 = ( ( 𝐷 𝑈 ) ( 𝑄 ( ( 𝑃 𝐷 ) 𝑊 ) ) )
cdleme43.v 𝑉 = ( ( 𝑍 𝑆 ) 𝑊 )
cdleme43.y 𝑌 = ( ( 𝑅 𝐷 ) 𝑊 )
Assertion cdleme43bN ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝐷𝐴 ∧ ¬ 𝐷 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdleme43.b 𝐵 = ( Base ‘ 𝐾 )
2 cdleme43.l = ( le ‘ 𝐾 )
3 cdleme43.j = ( join ‘ 𝐾 )
4 cdleme43.m = ( meet ‘ 𝐾 )
5 cdleme43.a 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme43.h 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme43.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
8 cdleme43.x 𝑋 = ( ( 𝑄 𝑃 ) 𝑊 )
9 cdleme43.c 𝐶 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
10 cdleme43.f 𝑍 = ( ( 𝑃 𝑄 ) ( 𝐶 ( ( 𝑅 𝑆 ) 𝑊 ) ) )
11 cdleme43.d 𝐷 = ( ( 𝑆 𝑋 ) ( 𝑃 ( ( 𝑄 𝑆 ) 𝑊 ) ) )
12 cdleme43.g 𝐺 = ( ( 𝑄 𝑃 ) ( 𝐷 ( ( 𝑍 𝑆 ) 𝑊 ) ) )
13 cdleme43.e 𝐸 = ( ( 𝐷 𝑈 ) ( 𝑄 ( ( 𝑃 𝐷 ) 𝑊 ) ) )
14 cdleme43.v 𝑉 = ( ( 𝑍 𝑆 ) 𝑊 )
15 cdleme43.y 𝑌 = ( ( 𝑅 𝐷 ) 𝑊 )
16 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
18 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
19 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
20 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → 𝑃𝑄 )
21 20 necomd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → 𝑄𝑃 )
22 simp3 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
23 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → 𝐾 ∈ HL )
24 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → 𝑃𝐴 )
25 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → 𝑄𝐴 )
26 3 5 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
27 23 24 25 26 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
28 27 breq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝑆 ( 𝑃 𝑄 ) ↔ 𝑆 ( 𝑄 𝑃 ) ) )
29 22 28 mtbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ¬ 𝑆 ( 𝑄 𝑃 ) )
30 2 3 4 5 6 8 11 cdleme3fa ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑄𝑃 ∧ ¬ 𝑆 ( 𝑄 𝑃 ) ) ) → 𝐷𝐴 )
31 16 17 18 19 21 29 30 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → 𝐷𝐴 )
32 2 3 4 5 6 8 11 cdleme3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑄𝑃 ∧ ¬ 𝑆 ( 𝑄 𝑃 ) ) ) → ¬ 𝐷 𝑊 )
33 16 17 18 19 21 29 32 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ¬ 𝐷 𝑊 )
34 31 33 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) → ( 𝐷𝐴 ∧ ¬ 𝐷 𝑊 ) )