Metamath Proof Explorer


Theorem cdleme21i

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 29-Nov-2012)

Ref Expression
Hypotheses cdleme21.l = ( le ‘ 𝐾 )
cdleme21.j = ( join ‘ 𝐾 )
cdleme21.m = ( meet ‘ 𝐾 )
cdleme21.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme21.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme21.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme21.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme21g.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
cdleme21g.d 𝐷 = ( ( 𝑅 𝑆 ) 𝑊 )
cdleme21g.y 𝑌 = ( ( 𝑅 𝑇 ) 𝑊 )
cdleme21g.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 𝐷 ) )
cdleme21g.o 𝑂 = ( ( 𝑃 𝑄 ) ( 𝐺 𝑌 ) )
Assertion cdleme21i ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → ( ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) → 𝑁 = 𝑂 ) )

Proof

Step Hyp Ref Expression
1 cdleme21.l = ( le ‘ 𝐾 )
2 cdleme21.j = ( join ‘ 𝐾 )
3 cdleme21.m = ( meet ‘ 𝐾 )
4 cdleme21.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme21.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme21.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme21.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
8 cdleme21g.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
9 cdleme21g.d 𝐷 = ( ( 𝑅 𝑆 ) 𝑊 )
10 cdleme21g.y 𝑌 = ( ( 𝑅 𝑇 ) 𝑊 )
11 cdleme21g.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 𝐷 ) )
12 cdleme21g.o 𝑂 = ( ( 𝑃 𝑄 ) ( 𝐺 𝑌 ) )
13 simpl11 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) ∧ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
15 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
16 simp21l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → 𝑆𝐴 )
17 14 15 16 3jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) )
18 17 adantr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) ∧ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) → ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) )
19 simp231 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → 𝑃𝑄 )
20 19 adantr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) ∧ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) → 𝑃𝑄 )
21 simp232 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
22 21 adantr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) ∧ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
23 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) ∧ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) → ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) )
24 1 2 4 5 4atexlem7 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ∃ 𝑧𝐴 ( ¬ 𝑧 𝑊 ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) )
25 13 18 20 22 23 24 syl113anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) ∧ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) → ∃ 𝑧𝐴 ( ¬ 𝑧 𝑊 ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) )
26 25 ex ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → ( ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) → ∃ 𝑧𝐴 ( ¬ 𝑧 𝑊 ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) ) )
27 1 2 3 4 5 6 7 8 9 10 11 12 cdleme21h ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → ( ∃ 𝑧𝐴 ( ¬ 𝑧 𝑊 ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) → 𝑁 = 𝑂 ) )
28 26 27 syld ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑈 ( 𝑆 𝑇 ) ) ) ) → ( ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) → 𝑁 = 𝑂 ) )