Metamath Proof Explorer

Theorem dalem42

Description: Lemma for dath . Auxiliary atoms G H I form a plane. (Contributed by NM, 4-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 ) ) ) )
dalem38.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem38.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem38.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem38.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem38.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
dalem38.h ⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) )
dalem38.i ⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) )
Assertion dalem42 ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 )

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 dalem38.m ⊢ ∧ = ( meet ‘ 𝐾 )
7 dalem38.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem38.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
9 dalem38.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
10 dalem38.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
11 dalem38.h ⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) )
12 dalem38.i ⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) )
13 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
14 13 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL )
15 1 2 3 4 5 6 7 8 9 10 dalem23 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ∈ 𝐴 )
16 1 2 3 4 5 6 7 8 9 11 dalem29 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐻 ∈ 𝐴 )
17 1 2 3 4 5 6 7 8 9 12 dalem34 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐼 ∈ 𝐴 )
18 1 2 3 4 5 6 7 8 9 10 11 12 dalem41 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ≠ 𝐻 )
19 1 2 3 4 5 6 7 8 9 10 11 12 dalem40 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝐼 ≤ ( 𝐺 ∨ 𝐻 ) )
20 2 3 4 7 lplni2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) ∧ ( 𝐺 ≠ 𝐻 ∧ ¬ 𝐼 ≤ ( 𝐺 ∨ 𝐻 ) ) ) → ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 )
21 14 15 16 17 18 19 20 syl132anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 )