Metamath Proof Explorer


Theorem cdleme16

Description: Part of proof of Lemma E in Crawley p. 113, conclusion of 3rd paragraph on p. 114. F and G represent f(s) and f(t) respectively. We show, in their notation, (s \/ t) /\ w=(f(s) \/ f(t)) /\ w, whether or not u <_ s \/ t. (Contributed by NM, 11-Oct-2012)

Ref Expression
Hypotheses cdleme12.l = ( le ‘ 𝐾 )
cdleme12.j = ( join ‘ 𝐾 )
cdleme12.m = ( meet ‘ 𝐾 )
cdleme12.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme12.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme12.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme12.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme12.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
Assertion cdleme16 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) → ( ( 𝑆 𝑇 ) 𝑊 ) = ( ( 𝐹 𝐺 ) 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdleme12.l = ( le ‘ 𝐾 )
2 cdleme12.j = ( join ‘ 𝐾 )
3 cdleme12.m = ( meet ‘ 𝐾 )
4 cdleme12.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme12.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme12.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme12.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
8 cdleme12.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
9 simpl11 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 simpl12 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
11 simpl13 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
12 simpl21 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
13 simpl22 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) )
14 simpl23 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑃𝑄𝑆𝑇 ) )
15 simpl3l ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
16 simpl3r ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ¬ 𝑇 ( 𝑃 𝑄 ) )
17 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → 𝑈 ( 𝑆 𝑇 ) )
18 1 2 3 4 5 6 7 8 cdleme11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝐹 𝐺 ) = ( 𝑆 𝑇 ) )
19 9 10 11 12 13 14 15 16 17 18 syl333anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝐹 𝐺 ) = ( 𝑆 𝑇 ) )
20 19 eqcomd ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑆 𝑇 ) = ( 𝐹 𝐺 ) )
21 20 oveq1d ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ 𝑈 ( 𝑆 𝑇 ) ) → ( ( 𝑆 𝑇 ) 𝑊 ) = ( ( 𝐹 𝐺 ) 𝑊 ) )
22 simpl11 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
23 simpl12 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
24 simpl13 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
25 simpl21 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
26 simpl22 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) )
27 simpl23 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ( 𝑃𝑄𝑆𝑇 ) )
28 simpl3l ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
29 simpl3r ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ¬ 𝑇 ( 𝑃 𝑄 ) )
30 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ¬ 𝑈 ( 𝑆 𝑇 ) )
31 1 2 3 4 5 6 7 8 cdleme16g ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑇 ) 𝑊 ) = ( ( 𝐹 𝐺 ) 𝑊 ) )
32 22 23 24 25 26 27 28 29 30 31 syl333anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) → ( ( 𝑆 𝑇 ) 𝑊 ) = ( ( 𝐹 𝐺 ) 𝑊 ) )
33 21 32 pm2.61dan ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) → ( ( 𝑆 𝑇 ) 𝑊 ) = ( ( 𝐹 𝐺 ) 𝑊 ) )