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Metamath Proof Explorer

Theorem dalem13

Description: Lemma for dalem14 . (Contributed by NM, 21-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 )
dalem13.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem13.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem13.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem13.w ⊢ 𝑊 = ( 𝑌 ∨ 𝐶 )
Assertion dalem13 ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∨ 𝑍 ) = 𝑊 )

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 dalem13.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6 dalem13.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
7 dalem13.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
8 dalem13.w ⊢ 𝑊 = ( 𝑌 ∨ 𝐶 )
9 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
10 9 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐾 ∈ HL )
11 1 dalemyeo ⊢ ( 𝜑 → 𝑌 ∈ 𝑂 )
12 11 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑌 ∈ 𝑂 )
13 1 dalemzeo ⊢ ( 𝜑 → 𝑍 ∈ 𝑂 )
14 13 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑍 ∈ 𝑂 )
15 eqid ⊢ ( LVols ‘ 𝐾 ) = ( LVols ‘ 𝐾 )
16 1 2 3 4 5 15 6 7 8 dalem9 ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑊 ∈ ( LVols ‘ 𝐾 ) )
17 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
18 1 5 dalemyeb ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
19 1 4 dalemceb ⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) )
20 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
21 20 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) → 𝑌 ≤ ( 𝑌 ∨ 𝐶 ) )
22 17 18 19 21 syl3anc ⊢ ( 𝜑 → 𝑌 ≤ ( 𝑌 ∨ 𝐶 ) )
23 22 8 breqtrrdi ⊢ ( 𝜑 → 𝑌 ≤ 𝑊 )
24 23 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑌 ≤ 𝑊 )
25 1 2 3 4 5 6 7 8 dalem8 ⊢ ( 𝜑 → 𝑍 ≤ 𝑊 )
26 25 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑍 ≤ 𝑊 )
27 simpr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑌 ≠ 𝑍 )
28 2 3 5 15 2lplnj ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ∧ 𝑊 ∈ ( LVols ‘ 𝐾 ) ) ∧ ( 𝑌 ≤ 𝑊 ∧ 𝑍 ≤ 𝑊 ∧ 𝑌 ≠ 𝑍 ) ) → ( 𝑌 ∨ 𝑍 ) = 𝑊 )
29 10 12 14 16 24 26 27 28 syl133anc ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∨ 𝑍 ) = 𝑊 )