Metamath Proof Explorer


Theorem dalem62

Description: Lemma for dath . Eliminate the condition ps containing dummy variables c and d . (Contributed by NM, 11-Aug-2012)

Ref Expression
Hypotheses dalem62.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 ) ) ) ) )
dalem62.l = ( le ‘ 𝐾 )
dalem62.j = ( join ‘ 𝐾 )
dalem62.a 𝐴 = ( Atoms ‘ 𝐾 )
dalem62.m = ( meet ‘ 𝐾 )
dalem62.o 𝑂 = ( LPlanes ‘ 𝐾 )
dalem62.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
dalem62.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
dalem62.d 𝐷 = ( ( 𝑃 𝑄 ) ( 𝑆 𝑇 ) )
dalem62.e 𝐸 = ( ( 𝑄 𝑅 ) ( 𝑇 𝑈 ) )
dalem62.f 𝐹 = ( ( 𝑅 𝑃 ) ( 𝑈 𝑆 ) )
Assertion dalem62 ( ( 𝜑𝑌 = 𝑍 ) → 𝐹 ( 𝐷 𝐸 ) )

Proof

Step Hyp Ref Expression
1 dalem62.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 dalem62.l = ( le ‘ 𝐾 )
3 dalem62.j = ( join ‘ 𝐾 )
4 dalem62.a 𝐴 = ( Atoms ‘ 𝐾 )
5 dalem62.m = ( meet ‘ 𝐾 )
6 dalem62.o 𝑂 = ( LPlanes ‘ 𝐾 )
7 dalem62.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
8 dalem62.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
9 dalem62.d 𝐷 = ( ( 𝑃 𝑄 ) ( 𝑆 𝑇 ) )
10 dalem62.e 𝐸 = ( ( 𝑄 𝑅 ) ( 𝑇 𝑈 ) )
11 dalem62.f 𝐹 = ( ( 𝑅 𝑃 ) ( 𝑈 𝑆 ) )
12 biid ( ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
13 1 2 3 4 12 6 7 8 dalem20 ( ( 𝜑𝑌 = 𝑍 ) → ∃ 𝑐𝑑 ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) )
14 1 2 3 4 12 5 6 7 8 9 10 11 dalem61 ( ( 𝜑𝑌 = 𝑍 ∧ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) ) → 𝐹 ( 𝐷 𝐸 ) )
15 14 3expia ( ( 𝜑𝑌 = 𝑍 ) → ( ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) → 𝐹 ( 𝐷 𝐸 ) ) )
16 15 exlimdvv ( ( 𝜑𝑌 = 𝑍 ) → ( ∃ 𝑐𝑑 ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 𝑐 𝑑 ) ) ) → 𝐹 ( 𝐷 𝐸 ) ) )
17 13 16 mpd ( ( 𝜑𝑌 = 𝑍 ) → 𝐹 ( 𝐷 𝐸 ) )