Metamath Proof Explorer

Theorem dalem52

Description: Lemma for dath . Lines G H and P Q intersect at an atom. (Contributed by NM, 8-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 ) ) ) )
dalem44.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem44.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem44.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem44.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem44.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
dalem44.h ⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) )
dalem44.i ⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) )
Assertion dalem52 ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 )

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 dalem44.m ⊢ ∧ = ( meet ‘ 𝐾 )
7 dalem44.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem44.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
9 dalem44.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
10 dalem44.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
11 dalem44.h ⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) )
12 dalem44.i ⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) )
13 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
14 13 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL )
15 5 4 dalemcceb ⊢ ( 𝜓 → 𝑐 ∈ ( Base ‘ 𝐾 ) )
16 15 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ ( Base ‘ 𝐾 ) )
17 14 16 jca ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐾 ∈ HL ∧ 𝑐 ∈ ( Base ‘ 𝐾 ) ) )
18 1 2 3 4 5 6 7 8 9 10 dalem23 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ∈ 𝐴 )
19 1 2 3 4 5 6 7 8 9 11 dalem29 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐻 ∈ 𝐴 )
20 1 2 3 4 5 6 7 8 9 12 dalem34 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐼 ∈ 𝐴 )
21 18 19 20 3jca ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) )
22 1 dalempea ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
23 1 dalemqea ⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
24 1 dalemrea ⊢ ( 𝜑 → 𝑅 ∈ 𝐴 )
25 22 23 24 3jca ⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) )
26 25 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) )
27 1 2 3 4 5 6 7 8 9 10 11 12 dalem42 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 )
28 1 dalemyeo ⊢ ( 𝜑 → 𝑌 ∈ 𝑂 )
29 28 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑌 ∈ 𝑂 )
30 1 2 3 4 5 6 7 8 9 10 11 12 dalem45 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) )
31 1 2 3 4 5 6 7 8 9 10 11 12 dalem46 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) )
32 1 2 3 4 5 6 7 8 9 10 11 12 dalem47 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) )
33 30 31 32 3jca ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ∧ ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ∧ ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 dalem48 ⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) )
35 1 2 3 4 5 6 7 8 9 10 11 12 dalem49 ⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 dalem50 ⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) )
37 34 35 36 3jca ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) )
38 37 3adant2 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) )
39 1 2 3 4 5 6 7 8 9 10 dalem27 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) )
40 1 2 3 4 5 6 7 8 9 11 dalem32 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) )
41 1 2 3 4 5 6 7 8 9 12 dalem36 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) )
42 39 40 41 3jca ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ∧ 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ∧ 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) )
43 biid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) ∧ ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂 ) ∧ ( ( ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ∧ ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ∧ ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) ∧ ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ∧ 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ∧ 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) ) ) ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) ∧ ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂 ) ∧ ( ( ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ∧ ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ∧ ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) ∧ ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ∧ 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ∧ 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) ) ) )
44 eqid ⊢ ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) = ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 )
45 eqid ⊢ ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) )
46 43 2 3 4 6 7 44 8 45 dalemdea ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) ∧ ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂 ) ∧ ( ( ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ∧ ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ∧ ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) ∧ ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ∧ 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ∧ 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) ) ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 )
47 17 21 26 27 29 33 38 42 46 syl323anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ∈ 𝐴 )