Metamath Proof Explorer

Theorem dalem60

Description: Lemma for dath . B is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012)

Ref Expression
Hypotheses dalem.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 ) ) ) ) )
dalem.l ⊢ ≤ = ( le ‘ 𝐾 )
dalem.j ⊢ ∨ = ( join ‘ 𝐾 )
dalem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dalem.ps ⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )
dalem60.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem60.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem60.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem60.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem60.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
dalem60.e ⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) )
dalem60.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
dalem60.h ⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) )
dalem60.i ⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) )
dalem60.b1 ⊢ 𝐵 = ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∧ 𝑌 )
Assertion dalem60 ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐷 ∨ 𝐸 ) = 𝐵 )

Proof

Step Hyp Ref Expression
1 dalem.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 dalem.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dalem.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dalem.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 dalem.ps ⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )
6 dalem60.m ⊢ ∧ = ( meet ‘ 𝐾 )
7 dalem60.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem60.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
9 dalem60.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
10 dalem60.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
11 dalem60.e ⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) )
12 dalem60.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
13 dalem60.h ⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) )
14 dalem60.i ⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) )
15 dalem60.b1 ⊢ 𝐵 = ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∧ 𝑌 )
16 1 2 3 4 5 6 7 8 9 10 12 13 14 15 dalem57 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐷 ≤ 𝐵 )
17 1 2 3 4 5 6 7 8 9 11 12 13 14 15 dalem58 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐸 ≤ 𝐵 )
18 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
19 18 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ Lat )
20 1 2 3 4 6 7 8 9 10 dalemdea ⊢ ( 𝜑 → 𝐷 ∈ 𝐴 )
21 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
22 21 4 atbase ⊢ ( 𝐷 ∈ 𝐴 → 𝐷 ∈ ( Base ‘ 𝐾 ) )
23 20 22 syl ⊢ ( 𝜑 → 𝐷 ∈ ( Base ‘ 𝐾 ) )
24 23 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐷 ∈ ( Base ‘ 𝐾 ) )
25 1 2 3 4 6 7 8 9 11 dalemeea ⊢ ( 𝜑 → 𝐸 ∈ 𝐴 )
26 21 4 atbase ⊢ ( 𝐸 ∈ 𝐴 → 𝐸 ∈ ( Base ‘ 𝐾 ) )
27 25 26 syl ⊢ ( 𝜑 → 𝐸 ∈ ( Base ‘ 𝐾 ) )
28 27 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐸 ∈ ( Base ‘ 𝐾 ) )
29 eqid ⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 )
30 1 2 3 4 5 6 29 7 8 9 12 13 14 15 dalem53 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐵 ∈ ( LLines ‘ 𝐾 ) )
31 21 29 llnbase ⊢ ( 𝐵 ∈ ( LLines ‘ 𝐾 ) → 𝐵 ∈ ( Base ‘ 𝐾 ) )
32 30 31 syl ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐵 ∈ ( Base ‘ 𝐾 ) )
33 21 2 3 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐷 ∈ ( Base ‘ 𝐾 ) ∧ 𝐸 ∈ ( Base ‘ 𝐾 ) ∧ 𝐵 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐷 ≤ 𝐵 ∧ 𝐸 ≤ 𝐵 ) ↔ ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ) )
34 19 24 28 32 33 syl13anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐷 ≤ 𝐵 ∧ 𝐸 ≤ 𝐵 ) ↔ ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ) )
35 16 17 34 mpbi2and ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 )
36 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
37 36 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL )
38 1 2 3 4 6 7 8 9 10 11 dalemdnee ⊢ ( 𝜑 → 𝐷 ≠ 𝐸 )
39 3 4 29 llni2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴 ) ∧ 𝐷 ≠ 𝐸 ) → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) )
40 36 20 25 38 39 syl31anc ⊢ ( 𝜑 → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) )
41 40 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) )
42 2 29 llncmp ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ∧ 𝐵 ∈ ( LLines ‘ 𝐾 ) ) → ( ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ↔ ( 𝐷 ∨ 𝐸 ) = 𝐵 ) )
43 37 41 30 42 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ↔ ( 𝐷 ∨ 𝐸 ) = 𝐵 ) )
44 35 43 mpbid ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐷 ∨ 𝐸 ) = 𝐵 )