Metamath Proof Explorer


Theorem dalem48

Description: Lemma for dath . Analogue of dalem45 for P Q . (Contributed by NM, 16-Aug-2012)

Ref Expression
Hypotheses dalem.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑆𝐴𝑇𝐴𝑈𝐴 ) ) ∧ ( 𝑌𝑂𝑍𝑂 ) ∧ ( ( ¬ 𝐶 ( 𝑃 𝑄 ) ∧ ¬ 𝐶 ( 𝑄 𝑅 ) ∧ ¬ 𝐶 ( 𝑅 𝑃 ) ) ∧ ( ¬ 𝐶 ( 𝑆 𝑇 ) ∧ ¬ 𝐶 ( 𝑇 𝑈 ) ∧ ¬ 𝐶 ( 𝑈 𝑆 ) ) ∧ ( 𝐶 ( 𝑃 𝑆 ) ∧ 𝐶 ( 𝑄 𝑇 ) ∧ 𝐶 ( 𝑅 𝑈 ) ) ) ) )
dalem.l = ( le ‘ 𝐾 )
dalem.j = ( join ‘ 𝐾 )
dalem.a 𝐴 = ( Atoms ‘ 𝐾 )
dalem.ps ( 𝜓 ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
dalem44.m = ( meet ‘ 𝐾 )
dalem44.o 𝑂 = ( LPlanes ‘ 𝐾 )
dalem44.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
dalem44.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
dalem44.g 𝐺 = ( ( 𝑐 𝑃 ) ( 𝑑 𝑆 ) )
dalem44.h 𝐻 = ( ( 𝑐 𝑄 ) ( 𝑑 𝑇 ) )
dalem44.i 𝐼 = ( ( 𝑐 𝑅 ) ( 𝑑 𝑈 ) )
Assertion dalem48 ( ( 𝜑𝜓 ) → ¬ 𝑐 ( 𝑃 𝑄 ) )

Proof

Step Hyp Ref Expression
1 dalem.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑆𝐴𝑇𝐴𝑈𝐴 ) ) ∧ ( 𝑌𝑂𝑍𝑂 ) ∧ ( ( ¬ 𝐶 ( 𝑃 𝑄 ) ∧ ¬ 𝐶 ( 𝑄 𝑅 ) ∧ ¬ 𝐶 ( 𝑅 𝑃 ) ) ∧ ( ¬ 𝐶 ( 𝑆 𝑇 ) ∧ ¬ 𝐶 ( 𝑇 𝑈 ) ∧ ¬ 𝐶 ( 𝑈 𝑆 ) ) ∧ ( 𝐶 ( 𝑃 𝑆 ) ∧ 𝐶 ( 𝑄 𝑇 ) ∧ 𝐶 ( 𝑅 𝑈 ) ) ) ) )
2 dalem.l = ( le ‘ 𝐾 )
3 dalem.j = ( join ‘ 𝐾 )
4 dalem.a 𝐴 = ( Atoms ‘ 𝐾 )
5 dalem.ps ( 𝜓 ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
6 dalem44.m = ( meet ‘ 𝐾 )
7 dalem44.o 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem44.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
9 dalem44.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
10 dalem44.g 𝐺 = ( ( 𝑐 𝑃 ) ( 𝑑 𝑆 ) )
11 dalem44.h 𝐻 = ( ( 𝑐 𝑄 ) ( 𝑑 𝑇 ) )
12 dalem44.i 𝐼 = ( ( 𝑐 𝑅 ) ( 𝑑 𝑈 ) )
13 1 dalemkelat ( 𝜑𝐾 ∈ Lat )
14 13 adantr ( ( 𝜑𝜓 ) → 𝐾 ∈ Lat )
15 5 4 dalemcceb ( 𝜓𝑐 ∈ ( Base ‘ 𝐾 ) )
16 15 adantl ( ( 𝜑𝜓 ) → 𝑐 ∈ ( Base ‘ 𝐾 ) )
17 1 3 4 dalempjqeb ( 𝜑 → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
18 17 adantr ( ( 𝜑𝜓 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
19 1 4 dalemreb ( 𝜑𝑅 ∈ ( Base ‘ 𝐾 ) )
20 19 adantr ( ( 𝜑𝜓 ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
21 5 dalem-ccly ( 𝜓 → ¬ 𝑐 𝑌 )
22 8 breq2i ( 𝑐 𝑌𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) )
23 21 22 sylnib ( 𝜓 → ¬ 𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) )
24 23 adantl ( ( 𝜑𝜓 ) → ¬ 𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) )
25 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
26 25 2 3 latnlej2l ( ( 𝐾 ∈ Lat ∧ ( 𝑐 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) ) → ¬ 𝑐 ( 𝑃 𝑄 ) )
27 14 16 18 20 24 26 syl131anc ( ( 𝜑𝜓 ) → ¬ 𝑐 ( 𝑃 𝑄 ) )