This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dalem5

Description: Lemma for dath . Atom U (in plane Z = S T U ) belongs to the 3-dimensional volume formed by Y and C . (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 )
dalem5.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem5.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem5.w ⊢ 𝑊 = ( 𝑌 ∨ 𝐶 )
Assertion dalem5 ( 𝜑 → 𝑈 ≤ 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 dalem5.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6 dalem5.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
7 dalem5.w ⊢ 𝑊 = ( 𝑌 ∨ 𝐶 )
8 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
10 1 4 dalemueb ⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
11 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
12 1 dalemrea ⊢ ( 𝜑 → 𝑅 ∈ 𝐴 )
13 1 2 3 4 5 6 dalemcea ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
14 8 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( 𝑅 ∨ 𝐶 ) ∈ ( Base ‘ 𝐾 ) )
15 11 12 13 14 syl3anc ⊢ ( 𝜑 → ( 𝑅 ∨ 𝐶 ) ∈ ( Base ‘ 𝐾 ) )
16 1 5 dalemyeb ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
17 1 4 dalemceb ⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) )
18 8 3 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑌 ∨ 𝐶 ) ∈ ( Base ‘ 𝐾 ) )
19 9 16 17 18 syl3anc ⊢ ( 𝜑 → ( 𝑌 ∨ 𝐶 ) ∈ ( Base ‘ 𝐾 ) )
20 7 19 eqeltrid ⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
21 1 dalemclrju ⊢ ( 𝜑 → 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) )
22 1 dalemuea ⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
23 1 dalempea ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
24 simp313 ⊢ ( ( ( ( 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 ) ) ) ) → ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) )
25 1 24 sylbi ⊢ ( 𝜑 → ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) )
26 2 3 4 atnlej1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) → 𝐶 ≠ 𝑅 )
27 11 13 12 23 25 26 syl131anc ⊢ ( 𝜑 → 𝐶 ≠ 𝑅 )
28 2 3 4 hlatexch1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐶 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝐶 ≠ 𝑅 ) → ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) ) )
29 11 13 22 12 27 28 syl131anc ⊢ ( 𝜑 → ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) ) )
30 21 29 mpd ⊢ ( 𝜑 → 𝑈 ≤ ( 𝑅 ∨ 𝐶 ) )
31 1 3 4 dalempjqeb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
32 1 4 dalemreb ⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
33 8 2 3 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → 𝑅 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
34 9 31 32 33 syl3anc ⊢ ( 𝜑 → 𝑅 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
35 34 6 breqtrrdi ⊢ ( 𝜑 → 𝑅 ≤ 𝑌 )
36 8 2 3 latjlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑅 ≤ 𝑌 → ( 𝑅 ∨ 𝐶 ) ≤ ( 𝑌 ∨ 𝐶 ) ) )
37 9 32 16 17 36 syl13anc ⊢ ( 𝜑 → ( 𝑅 ≤ 𝑌 → ( 𝑅 ∨ 𝐶 ) ≤ ( 𝑌 ∨ 𝐶 ) ) )
38 35 37 mpd ⊢ ( 𝜑 → ( 𝑅 ∨ 𝐶 ) ≤ ( 𝑌 ∨ 𝐶 ) )
39 38 7 breqtrrdi ⊢ ( 𝜑 → ( 𝑅 ∨ 𝐶 ) ≤ 𝑊 )
40 8 2 9 10 15 20 30 39 lattrd ⊢ ( 𝜑 → 𝑈 ≤ 𝑊 )