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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 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 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 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 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 ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) )

Proof

Step Hyp Ref Expression
1 dalem.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 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 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 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 ⊢ ( 𝜑 → 𝐾 ∈ Lat )
14 13 adantr ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐾 ∈ Lat )
15 5 4 dalemcceb ⊢ ( 𝜓 → 𝑐 ∈ ( Base ‘ 𝐾 ) )
16 15 adantl ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑐 ∈ ( Base ‘ 𝐾 ) )
17 1 3 4 dalempjqeb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
18 17 adantr ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
19 1 4 dalemreb ⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
20 19 adantr ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
21 5 dalem-ccly ⊢ ( 𝜓 → ¬ 𝑐 ≤ 𝑌 )
22 8 breq2i ⊢ ( 𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
23 21 22 sylnib ⊢ ( 𝜓 → ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
24 23 adantl ⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 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 ⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) )