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Metamath Proof Explorer

Theorem dalem10

Description: Lemma for dath . Atom D belongs to the axis of perspectivity X . (Contributed by NM, 19-Jul-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 )
dalem10.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem10.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem10.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem10.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem10.x ⊢ 𝑋 = ( 𝑌 ∧ 𝑍 )
dalem10.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
Assertion dalem10 ( 𝜑 → 𝐷 ≤ 𝑋 )

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 dalem10.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dalem10.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
7 dalem10.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
8 dalem10.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
9 dalem10.x ⊢ 𝑋 = ( 𝑌 ∧ 𝑍 )
10 dalem10.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
11 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
12 1 3 4 dalempjqeb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
13 1 4 dalemreb ⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
14 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
15 14 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
16 11 12 13 15 syl3anc ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
17 1 3 4 dalemsjteb ⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
18 1 4 dalemueb ⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
19 14 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
20 11 17 18 19 syl3anc ⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
21 1 6 dalemyeb ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
22 7 21 eqeltrrid ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
23 1 dalemzeo ⊢ ( 𝜑 → 𝑍 ∈ 𝑂 )
24 14 6 lplnbase ⊢ ( 𝑍 ∈ 𝑂 → 𝑍 ∈ ( Base ‘ 𝐾 ) )
25 23 24 syl ⊢ ( 𝜑 → 𝑍 ∈ ( Base ‘ 𝐾 ) )
26 8 25 eqeltrrid ⊢ ( 𝜑 → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
27 14 2 5 latmlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) ) )
28 11 12 22 17 26 27 syl122anc ⊢ ( 𝜑 → ( ( ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) ) )
29 16 20 28 mp2and ⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) )
30 7 8 oveq12i ⊢ ( 𝑌 ∧ 𝑍 ) = ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
31 9 30 eqtri ⊢ 𝑋 = ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
32 29 10 31 3brtr4g ⊢ ( 𝜑 → 𝐷 ≤ 𝑋 )