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Metamath Proof Explorer

Theorem dalem16

Description: Lemma for dath . The atoms D , E , and F form a line of perspectivity. This is Desargues's theorem for the special case where planes Y and Z are different. (Contributed by NM, 7-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 )
dalem16.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem16.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem16.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem16.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem16.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
dalem16.e ⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) )
dalem16.f ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑈 ∨ 𝑆 ) )
Assertion dalem16 ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐹 ≤ ( 𝐷 ∨ 𝐸 ) )

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 dalem16.m ⊢ ∧ = ( meet ‘ 𝐾 )
6 dalem16.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
7 dalem16.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
8 dalem16.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
9 dalem16.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
10 dalem16.e ⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) )
11 dalem16.f ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑈 ∨ 𝑆 ) )
12 eqid ⊢ ( 𝑌 ∧ 𝑍 ) = ( 𝑌 ∧ 𝑍 )
13 1 2 3 4 5 6 7 8 12 11 dalem12 ⊢ ( 𝜑 → 𝐹 ≤ ( 𝑌 ∧ 𝑍 ) )
14 13 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐹 ≤ ( 𝑌 ∧ 𝑍 ) )
15 1 2 3 4 5 6 7 8 12 9 dalem10 ⊢ ( 𝜑 → 𝐷 ≤ ( 𝑌 ∧ 𝑍 ) )
16 1 2 3 4 5 6 7 8 12 10 dalem11 ⊢ ( 𝜑 → 𝐸 ≤ ( 𝑌 ∧ 𝑍 ) )
17 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
18 1 2 3 4 5 6 7 8 9 dalemdea ⊢ ( 𝜑 → 𝐷 ∈ 𝐴 )
19 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
20 19 4 atbase ⊢ ( 𝐷 ∈ 𝐴 → 𝐷 ∈ ( Base ‘ 𝐾 ) )
21 18 20 syl ⊢ ( 𝜑 → 𝐷 ∈ ( Base ‘ 𝐾 ) )
22 1 2 3 4 5 6 7 8 10 dalemeea ⊢ ( 𝜑 → 𝐸 ∈ 𝐴 )
23 19 4 atbase ⊢ ( 𝐸 ∈ 𝐴 → 𝐸 ∈ ( Base ‘ 𝐾 ) )
24 22 23 syl ⊢ ( 𝜑 → 𝐸 ∈ ( Base ‘ 𝐾 ) )
25 1 6 dalemyeb ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
26 1 dalemzeo ⊢ ( 𝜑 → 𝑍 ∈ 𝑂 )
27 19 6 lplnbase ⊢ ( 𝑍 ∈ 𝑂 → 𝑍 ∈ ( Base ‘ 𝐾 ) )
28 26 27 syl ⊢ ( 𝜑 → 𝑍 ∈ ( Base ‘ 𝐾 ) )
29 19 5 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝑍 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑌 ∧ 𝑍 ) ∈ ( Base ‘ 𝐾 ) )
30 17 25 28 29 syl3anc ⊢ ( 𝜑 → ( 𝑌 ∧ 𝑍 ) ∈ ( Base ‘ 𝐾 ) )
31 19 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐷 ∈ ( Base ‘ 𝐾 ) ∧ 𝐸 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑌 ∧ 𝑍 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐷 ≤ ( 𝑌 ∧ 𝑍 ) ∧ 𝐸 ≤ ( 𝑌 ∧ 𝑍 ) ) ↔ ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ) )
32 17 21 24 30 31 syl13anc ⊢ ( 𝜑 → ( ( 𝐷 ≤ ( 𝑌 ∧ 𝑍 ) ∧ 𝐸 ≤ ( 𝑌 ∧ 𝑍 ) ) ↔ ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ) )
33 15 16 32 mpbi2and ⊢ ( 𝜑 → ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) )
34 33 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) )
35 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
36 35 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐾 ∈ HL )
37 1 2 3 4 5 6 7 8 9 10 dalemdnee ⊢ ( 𝜑 → 𝐷 ≠ 𝐸 )
38 eqid ⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 )
39 3 4 38 llni2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴 ) ∧ 𝐷 ≠ 𝐸 ) → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) )
40 35 18 22 37 39 syl31anc ⊢ ( 𝜑 → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) )
41 40 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) )
42 1 2 3 4 5 38 6 7 8 12 dalem15 ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∧ 𝑍 ) ∈ ( LLines ‘ 𝐾 ) )
43 2 38 llncmp ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ∧ ( 𝑌 ∧ 𝑍 ) ∈ ( LLines ‘ 𝐾 ) ) → ( ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ↔ ( 𝐷 ∨ 𝐸 ) = ( 𝑌 ∧ 𝑍 ) ) )
44 36 41 42 43 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ↔ ( 𝐷 ∨ 𝐸 ) = ( 𝑌 ∧ 𝑍 ) ) )
45 34 44 mpbid ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝐷 ∨ 𝐸 ) = ( 𝑌 ∧ 𝑍 ) )
46 14 45 breqtrrd ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐹 ≤ ( 𝐷 ∨ 𝐸 ) )