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 ) ) ∧ ( 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 ) ) ) ) )
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 ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) )

Proof

Step Hyp Ref Expression
1 dalema.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 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 ( 𝜑𝐾 ∈ HL )
9 8 ad3antrrr ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → 𝐾 ∈ HL )
10 1 2 3 4 5 6 dalemcea ( 𝜑𝐶𝐴 )
11 10 ad3antrrr ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → 𝐶𝐴 )
12 simplr ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → 𝑐𝐴 )
13 1 5 dalemyeb ( 𝜑𝑌 ∈ ( Base ‘ 𝐾 ) )
14 13 ad3antrrr ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → 𝑌 ∈ ( Base ‘ 𝐾 ) )
15 1 2 3 4 5 6 7 dalem17 ( ( 𝜑𝑌 = 𝑍 ) → 𝐶 𝑌 )
16 15 ad2antrr ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → 𝐶 𝑌 )
17 simpr ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → ¬ 𝑐 𝑌 )
18 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
19 18 2 3 4 atbtwnex ( ( ( 𝐾 ∈ HL ∧ 𝐶𝐴𝑐𝐴 ) ∧ ( 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 𝑌 ∧ ¬ 𝑐 𝑌 ) ) → ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) )
20 9 11 12 14 16 17 19 syl33anc ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) )