Metamath Proof Explorer


Theorem dalem18

Description: Lemma for dath . Show that a dummy atom c exists outside of the Y and Z planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargues's theorem does not always hold in 2 dimensions.) (Contributed by NM, 29-Jul-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 )
dalem18.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
Assertion dalem18 ( 𝜑 → ∃ 𝑐𝐴 ¬ 𝑐 𝑌 )

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 dalem18.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
6 1 dalemkehl ( 𝜑𝐾 ∈ HL )
7 1 dalempea ( 𝜑𝑃𝐴 )
8 1 dalemqea ( 𝜑𝑄𝐴 )
9 1 dalemrea ( 𝜑𝑅𝐴 )
10 3 2 4 3dim3 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → ∃ 𝑐𝐴 ¬ 𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) )
11 6 7 8 9 10 syl13anc ( 𝜑 → ∃ 𝑐𝐴 ¬ 𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) )
12 5 breq2i ( 𝑐 𝑌𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) )
13 12 notbii ( ¬ 𝑐 𝑌 ↔ ¬ 𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) )
14 13 rexbii ( ∃ 𝑐𝐴 ¬ 𝑐 𝑌 ↔ ∃ 𝑐𝐴 ¬ 𝑐 ( ( 𝑃 𝑄 ) 𝑅 ) )
15 11 14 sylibr ( 𝜑 → ∃ 𝑐𝐴 ¬ 𝑐 𝑌 )