Metamath Proof Explorer


Theorem cdleme18a

Description: Part of proof of Lemma E in Crawley p. 114, 2nd sentence of 4th paragraph. F , G represent f(s), f_s(q) respectively. We show -. f_s(q) <_ w. (Contributed by NM, 12-Oct-2012)

Ref Expression
Hypotheses cdleme18.l = ( le ‘ 𝐾 )
cdleme18.j = ( join ‘ 𝐾 )
cdleme18.m = ( meet ‘ 𝐾 )
cdleme18.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme18.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme18.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme18.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme18.g 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑄 𝑆 ) 𝑊 ) ) )
Assertion cdleme18a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐺 𝑊 )

Proof

Step Hyp Ref Expression
1 cdleme18.l = ( le ‘ 𝐾 )
2 cdleme18.j = ( join ‘ 𝐾 )
3 cdleme18.m = ( meet ‘ 𝐾 )
4 cdleme18.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme18.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme18.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme18.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
8 cdleme18.g 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑄 𝑆 ) 𝑊 ) ) )
9 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 simp21 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
11 simp22 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
12 simp23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
13 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑃𝑄 )
14 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ HL )
15 simp21l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑃𝐴 )
16 simp22l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑄𝐴 )
17 1 2 4 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑄 ( 𝑃 𝑄 ) )
18 14 15 16 17 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑄 ( 𝑃 𝑄 ) )
19 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
20 1 2 3 4 5 6 7 8 cdleme7 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑄 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐺 𝑊 )
21 9 10 11 11 12 13 18 19 20 syl323anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐺 𝑊 )