Metamath Proof Explorer


Theorem cdleme21f

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 ) ) )
cdleme21.b 𝐵 = ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) )
cdleme21.d 𝐷 = ( ( 𝑅 𝑆 ) 𝑊 )
cdleme21.e 𝐸 = ( ( 𝑅 𝑧 ) 𝑊 )
cdleme21d.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 𝐷 ) )
cdleme21d.z 𝑍 = ( ( 𝑃 𝑄 ) ( 𝐵 𝐸 ) )
cdleme21.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
cdleme21.y 𝑌 = ( ( 𝑅 𝑇 ) 𝑊 )
cdleme21.o 𝑂 = ( ( 𝑃 𝑄 ) ( 𝐺 𝑌 ) )
Assertion cdleme21f ( ( ( ( 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 )

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 cdleme21.b 𝐵 = ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) )
9 cdleme21.d 𝐷 = ( ( 𝑅 𝑆 ) 𝑊 )
10 cdleme21.e 𝐸 = ( ( 𝑅 𝑧 ) 𝑊 )
11 cdleme21d.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 𝐷 ) )
12 cdleme21d.z 𝑍 = ( ( 𝑃 𝑄 ) ( 𝐵 𝐸 ) )
13 cdleme21.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
14 cdleme21.y 𝑌 = ( ( 𝑅 𝑇 ) 𝑊 )
15 cdleme21.o 𝑂 = ( ( 𝑃 𝑄 ) ( 𝐺 𝑌 ) )
16 simp11 ( ( ( ( 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 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 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 ) ) ∧ ( ( 𝑧𝐴 ∧ ¬ 𝑧 𝑊 ) ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
18 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 ) ) ∧ ( ( 𝑧𝐴 ∧ ¬ 𝑧 𝑊 ) ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
19 simp31 ( ( ( ( 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 ) ) ) ) → ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) )
20 simp21 ( ( ( ( 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 ) ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
21 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 ) ) ∧ ( ( 𝑧𝐴 ∧ ¬ 𝑧 𝑊 ) ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) ) ) → 𝑃𝑄 )
22 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 ) ) ∧ ( ( 𝑧𝐴 ∧ ¬ 𝑧 𝑊 ) ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
23 simp32l ( ( ( ( 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 ) ) ) ) → 𝑅 ( 𝑃 𝑄 ) )
24 22 23 jca ( ( ( ( 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 ) ) ) ) → ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) )
25 simp33 ( ( ( ( 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 ) ) ) ) → ( ( 𝑧𝐴 ∧ ¬ 𝑧 𝑊 ) ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) )
26 1 2 3 4 5 6 7 8 9 10 11 12 cdleme21d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ∧ ( ( 𝑧𝐴 ∧ ¬ 𝑧 𝑊 ) ∧ ( 𝑃 𝑧 ) = ( 𝑆 𝑧 ) ) ) ) → 𝑁 = 𝑍 )
27 16 17 18 19 20 21 24 25 26 syl323anc ( ( ( ( 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 = 𝑍 )
28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 cdleme21e ( ( ( ( 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 ) ) ) ) → 𝑂 = 𝑍 )
29 27 28 eqtr4d ( ( ( ( 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 )