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 ) ) ∧ ( 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 ) ) ) ) )
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 ( 𝜑𝑈 𝑊 )

Proof

Step Hyp Ref Expression
1 dalema.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 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 ( 𝜑𝐾 ∈ Lat )
10 1 4 dalemueb ( 𝜑𝑈 ∈ ( Base ‘ 𝐾 ) )
11 1 dalemkehl ( 𝜑𝐾 ∈ HL )
12 1 dalemrea ( 𝜑𝑅𝐴 )
13 1 2 3 4 5 6 dalemcea ( 𝜑𝐶𝐴 )
14 8 3 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑅𝐴𝐶𝐴 ) → ( 𝑅 𝐶 ) ∈ ( Base ‘ 𝐾 ) )
15 11 12 13 14 syl3anc ( 𝜑 → ( 𝑅 𝐶 ) ∈ ( Base ‘ 𝐾 ) )
16 1 5 dalemyeb ( 𝜑𝑌 ∈ ( Base ‘ 𝐾 ) )
17 1 4 dalemceb ( 𝜑𝐶 ∈ ( 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 ( 𝜑𝑊 ∈ ( Base ‘ 𝐾 ) )
21 1 dalemclrju ( 𝜑𝐶 ( 𝑅 𝑈 ) )
22 1 dalemuea ( 𝜑𝑈𝐴 )
23 1 dalempea ( 𝜑𝑃𝐴 )
24 simp313 ( ( ( ( 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 ) ) ) ) → ¬ 𝐶 ( 𝑅 𝑃 ) )
25 1 24 sylbi ( 𝜑 → ¬ 𝐶 ( 𝑅 𝑃 ) )
26 2 3 4 atnlej1 ( ( 𝐾 ∈ HL ∧ ( 𝐶𝐴𝑅𝐴𝑃𝐴 ) ∧ ¬ 𝐶 ( 𝑅 𝑃 ) ) → 𝐶𝑅 )
27 11 13 12 23 25 26 syl131anc ( 𝜑𝐶𝑅 )
28 2 3 4 hlatexch1 ( ( 𝐾 ∈ HL ∧ ( 𝐶𝐴𝑈𝐴𝑅𝐴 ) ∧ 𝐶𝑅 ) → ( 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 ( 𝜑𝑈 ( 𝑅 𝐶 ) )
31 1 3 4 dalempjqeb ( 𝜑 → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
32 1 4 dalemreb ( 𝜑𝑅 ∈ ( 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 ( 𝜑𝑅 ( ( 𝑃 𝑄 ) 𝑅 ) )
35 34 6 breqtrrdi ( 𝜑𝑅 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 ( 𝜑𝑈 𝑊 )