Metamath Proof Explorer


Theorem cdleme18c

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) = p whenever p \/ q has three atoms under it (implied by the negated existential condition). (Contributed by NM, 10-Nov-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 cdleme18c ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐺 = 𝑃 )

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 simp31 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑃𝑄 )
10 simp32 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
11 9 10 jca ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) )
12 1 2 3 4 5 6 7 8 cdleme18b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐺𝑄 )
13 11 12 syld3an3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐺𝑄 )
14 13 neneqd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ¬ 𝐺 = 𝑄 )
15 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐾 ∈ HL )
16 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑊𝐻 )
17 simp21l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑃𝐴 )
18 simp22l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑄𝐴 )
19 simp23l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑆𝐴 )
20 1 2 3 4 5 6 7 8 cdleme4a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴𝑄𝐴 ) ∧ 𝑆𝐴 ) → 𝐺 ( 𝑃 𝑄 ) )
21 15 16 17 18 18 19 20 syl231anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐺 ( 𝑃 𝑄 ) )
22 simp33 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) )
23 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
24 simp21 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
25 simp22 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
26 simp23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
27 1 2 4 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑄 ( 𝑃 𝑄 ) )
28 15 17 18 27 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑄 ( 𝑃 𝑄 ) )
29 1 2 3 4 5 6 7 8 cdleme7ga ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑄 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐺𝐴 )
30 23 24 25 25 26 9 28 10 29 syl323anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐺𝐴 )
31 1 2 3 4 5 6 7 8 cdleme18a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐺 𝑊 )
32 11 31 syld3an3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ¬ 𝐺 𝑊 )
33 1 2 4 cdleme0nex ( ( ( 𝐾 ∈ HL ∧ 𝐺 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ∧ ( 𝑃𝐴𝑄𝐴𝑃𝑄 ) ∧ ( 𝐺𝐴 ∧ ¬ 𝐺 𝑊 ) ) → ( 𝐺 = 𝑃𝐺 = 𝑄 ) )
34 15 21 22 17 18 9 30 32 33 syl332anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺 = 𝑃𝐺 = 𝑄 ) )
35 34 ord ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ¬ 𝐺 = 𝑃𝐺 = 𝑄 ) )
36 14 35 mt3d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐺 = 𝑃 )