Metamath Proof Explorer


Theorem dalem20

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, 14-Aug-2012)

Ref Expression
Hypotheses dalem.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 ) ) ) ) )
dalem.l = ( le ‘ 𝐾 )
dalem.j = ( join ‘ 𝐾 )
dalem.a 𝐴 = ( Atoms ‘ 𝐾 )
dalem.ps ( 𝜓 ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
dalem20.o 𝑂 = ( LPlanes ‘ 𝐾 )
dalem20.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
dalem20.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
Assertion dalem20 ( ( 𝜑𝑌 = 𝑍 ) → ∃ 𝑐𝑑 𝜓 )

Proof

Step Hyp Ref Expression
1 dalem.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 dalem.l = ( le ‘ 𝐾 )
3 dalem.j = ( join ‘ 𝐾 )
4 dalem.a 𝐴 = ( Atoms ‘ 𝐾 )
5 dalem.ps ( 𝜓 ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
6 dalem20.o 𝑂 = ( LPlanes ‘ 𝐾 )
7 dalem20.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
8 dalem20.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
9 1 2 3 4 7 dalem18 ( 𝜑 → ∃ 𝑐𝐴 ¬ 𝑐 𝑌 )
10 9 adantr ( ( 𝜑𝑌 = 𝑍 ) → ∃ 𝑐𝐴 ¬ 𝑐 𝑌 )
11 1 2 3 4 6 7 8 dalem19 ( ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) ∧ ¬ 𝑐 𝑌 ) → ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) )
12 11 ex ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) → ( ¬ 𝑐 𝑌 → ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
13 12 ancld ( ( ( 𝜑𝑌 = 𝑍 ) ∧ 𝑐𝐴 ) → ( ¬ 𝑐 𝑌 → ( ¬ 𝑐 𝑌 ∧ ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ) )
14 13 reximdva ( ( 𝜑𝑌 = 𝑍 ) → ( ∃ 𝑐𝐴 ¬ 𝑐 𝑌 → ∃ 𝑐𝐴 ( ¬ 𝑐 𝑌 ∧ ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ) )
15 10 14 mpd ( ( 𝜑𝑌 = 𝑍 ) → ∃ 𝑐𝐴 ( ¬ 𝑐 𝑌 ∧ ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
16 3anass ( ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ( ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ) )
17 5 16 bitri ( 𝜓 ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ( ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ) )
18 17 2exbii ( ∃ 𝑐𝑑 𝜓 ↔ ∃ 𝑐𝑑 ( ( 𝑐𝐴𝑑𝐴 ) ∧ ( ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ) )
19 r2ex ( ∃ 𝑐𝐴𝑑𝐴 ( ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ↔ ∃ 𝑐𝑑 ( ( 𝑐𝐴𝑑𝐴 ) ∧ ( ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ) )
20 r19.42v ( ∃ 𝑑𝐴 ( ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ↔ ( ¬ 𝑐 𝑌 ∧ ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
21 20 rexbii ( ∃ 𝑐𝐴𝑑𝐴 ( ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ↔ ∃ 𝑐𝐴 ( ¬ 𝑐 𝑌 ∧ ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
22 18 19 21 3bitr2ri ( ∃ 𝑐𝐴 ( ¬ 𝑐 𝑌 ∧ ∃ 𝑑𝐴 ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ↔ ∃ 𝑐𝑑 𝜓 )
23 15 22 sylib ( ( 𝜑𝑌 = 𝑍 ) → ∃ 𝑐𝑑 𝜓 )