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Metamath Proof Explorer

Theorem dalem17

Description: Lemma for dath . When planes Y and Z are equal, the center of perspectivity C is in Y . (Contributed by NM, 1-Aug-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 )
dalem17.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem17.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem17.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
Assertion dalem17 ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝐶 ≤ 𝑌 )

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 dalem17.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6 dalem17.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
7 dalem17.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
8 1 dalemclrju ⊢ ( 𝜑 → 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) )
9 8 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) )
10 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
11 1 3 4 dalempjqeb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
12 1 4 dalemreb ⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
13 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
14 13 2 3 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → 𝑅 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
15 10 11 12 14 syl3anc ⊢ ( 𝜑 → 𝑅 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
16 15 6 breqtrrdi ⊢ ( 𝜑 → 𝑅 ≤ 𝑌 )
17 16 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑅 ≤ 𝑌 )
18 1 3 4 dalemsjteb ⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
19 1 4 dalemueb ⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
20 13 2 3 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → 𝑈 ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
21 10 18 19 20 syl3anc ⊢ ( 𝜑 → 𝑈 ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
22 21 7 breqtrrdi ⊢ ( 𝜑 → 𝑈 ≤ 𝑍 )
23 22 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑈 ≤ 𝑍 )
24 simpr ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑌 = 𝑍 )
25 23 24 breqtrrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑈 ≤ 𝑌 )
26 1 5 dalemyeb ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
27 13 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌 ) ↔ ( 𝑅 ∨ 𝑈 ) ≤ 𝑌 ) )
28 10 12 19 26 27 syl13anc ⊢ ( 𝜑 → ( ( 𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌 ) ↔ ( 𝑅 ∨ 𝑈 ) ≤ 𝑌 ) )
29 28 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( ( 𝑅 ≤ 𝑌 ∧ 𝑈 ≤ 𝑌 ) ↔ ( 𝑅 ∨ 𝑈 ) ≤ 𝑌 ) )
30 17 25 29 mpbi2and ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( 𝑅 ∨ 𝑈 ) ≤ 𝑌 )
31 1 4 dalemceb ⊢ ( 𝜑 → 𝐶 ∈ ( Base ‘ 𝐾 ) )
32 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
33 1 dalemrea ⊢ ( 𝜑 → 𝑅 ∈ 𝐴 )
34 1 dalemuea ⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
35 13 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) → ( 𝑅 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
36 32 33 34 35 syl3anc ⊢ ( 𝜑 → ( 𝑅 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
37 13 2 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐶 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑅 ∨ 𝑈 ) ≤ 𝑌 ) → 𝐶 ≤ 𝑌 ) )
38 10 31 36 26 37 syl13anc ⊢ ( 𝜑 → ( ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑅 ∨ 𝑈 ) ≤ 𝑌 ) → 𝐶 ≤ 𝑌 ) )
39 38 adantr ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( ( 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑅 ∨ 𝑈 ) ≤ 𝑌 ) → 𝐶 ≤ 𝑌 ) )
40 9 30 39 mp2and ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝐶 ≤ 𝑌 )