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Metamath Proof Explorer

Theorem dalem19

Description: Lemma for dath . Show that a second dummy atom d exists outside of the Y and Z planes (when those planes are equal). (Contributed by NM, 15-Aug-2012)

Ref Expression
Hypotheses dalema.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 ) ) ) ) )
dalemc.l ⊢ ≤ = ( le ‘ 𝐾 )
dalemc.j ⊢ ∨ = ( join ‘ 𝐾 )
dalemc.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dalem19.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem19.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem19.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
Assertion dalem19 ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) )

Proof

Step Hyp Ref Expression
1 dalema.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 dalemc.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dalemc.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dalemc.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 dalem19.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6 dalem19.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
7 dalem19.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
8 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
9 8 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝐾 ∈ HL )
10 1 2 3 4 5 6 dalemcea ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
11 10 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝐶 ∈ 𝐴 )
12 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝑐 ∈ 𝐴 )
13 1 5 dalemyeb ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
14 13 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝑌 ∈ ( Base ‘ 𝐾 ) )
15 1 2 3 4 5 6 7 dalem17 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝐶 ≤ 𝑌 )
16 15 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝐶 ≤ 𝑌 )
17 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → ¬ 𝑐 ≤ 𝑌 )
18 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
19 18 2 3 4 atbtwnex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ) ∧ ( 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ≤ 𝑌 ∧ ¬ 𝑐 ≤ 𝑌 ) ) → ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) )
20 9 11 12 14 16 17 19 syl33anc ⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) )