Metamath Proof Explorer


Theorem cdleme21g

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 cdleme21g ( ( ( ( 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 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 eqid ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) ) = ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) )
14 eqid ( ( 𝑅 𝑧 ) 𝑊 ) = ( ( 𝑅 𝑧 ) 𝑊 )
15 eqid ( ( 𝑃 𝑄 ) ( ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) ) ( ( 𝑅 𝑧 ) 𝑊 ) ) ) = ( ( 𝑃 𝑄 ) ( ( ( 𝑧 𝑈 ) ( 𝑄 ( ( 𝑃 𝑧 ) 𝑊 ) ) ) ( ( 𝑅 𝑧 ) 𝑊 ) ) )
16 1 2 3 4 5 6 7 13 9 14 11 15 8 10 12 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 )