Metamath Proof Explorer

Theorem dalem24

Description: Lemma for dath . Show that auxiliary atom G is outside of plane Y . (Contributed by NM, 2-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 ) ) ) )
dalem23.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem23.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem23.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem23.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem23.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
Assertion dalem24 ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝐺 ≤ 𝑌 )

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 dalem23.m ⊢ ∧ = ( meet ‘ 𝐾 )
7 dalem23.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem23.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
9 dalem23.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
10 dalem23.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
11 10 oveq1i ⊢ ( 𝐺 ∧ 𝑌 ) = ( ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ∧ 𝑌 )
12 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
13 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
14 12 13 syl ⊢ ( 𝜑 → 𝐾 ∈ OL )
15 14 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ OL )
16 12 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL )
17 5 dalemccea ⊢ ( 𝜓 → 𝑐 ∈ 𝐴 )
18 17 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ 𝐴 )
19 1 dalempea ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
20 19 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ∈ 𝐴 )
21 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
22 21 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
23 16 18 20 22 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
24 5 dalemddea ⊢ ( 𝜓 → 𝑑 ∈ 𝐴 )
25 24 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑑 ∈ 𝐴 )
26 1 dalemsea ⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
27 26 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑆 ∈ 𝐴 )
28 21 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
29 16 25 27 28 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
30 1 7 dalemyeb ⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
31 30 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑌 ∈ ( Base ‘ 𝐾 ) )
32 21 6 latmmdir ⊢ ( ( 𝐾 ∈ OL ∧ ( ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ∧ 𝑌 ) = ( ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) ∧ ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) ) )
33 15 23 29 31 32 syl13anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ∧ 𝑌 ) = ( ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) ∧ ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) ) )
34 11 33 syl5eq ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ∧ 𝑌 ) = ( ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) ∧ ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) ) )
35 3 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑃 ) = ( 𝑃 ∨ 𝑐 ) )
36 16 18 20 35 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑃 ) = ( 𝑃 ∨ 𝑐 ) )
37 36 oveq1d ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) = ( ( 𝑃 ∨ 𝑐 ) ∧ 𝑌 ) )
38 1 2 3 4 7 8 dalemply ⊢ ( 𝜑 → 𝑃 ≤ 𝑌 )
39 38 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ≤ 𝑌 )
40 5 dalem-ccly ⊢ ( 𝜓 → ¬ 𝑐 ≤ 𝑌 )
41 40 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ 𝑌 )
42 21 2 3 6 4 2atjm ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ≤ 𝑌 ∧ ¬ 𝑐 ≤ 𝑌 ) ) → ( ( 𝑃 ∨ 𝑐 ) ∧ 𝑌 ) = 𝑃 )
43 16 20 18 31 39 41 42 syl132anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑃 ∨ 𝑐 ) ∧ 𝑌 ) = 𝑃 )
44 37 43 eqtrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) = 𝑃 )
45 3 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑑 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑑 ) )
46 16 25 27 45 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑑 ) )
47 46 oveq1d ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) = ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) )
48 1 2 3 4 9 dalemsly ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑆 ≤ 𝑌 )
49 48 3adant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑆 ≤ 𝑌 )
50 5 dalem-ddly ⊢ ( 𝜓 → ¬ 𝑑 ≤ 𝑌 )
51 50 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑑 ≤ 𝑌 )
52 21 2 3 6 4 2atjm ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑆 ≤ 𝑌 ∧ ¬ 𝑑 ≤ 𝑌 ) ) → ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) = 𝑆 )
53 16 27 25 31 49 51 52 syl132anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) = 𝑆 )
54 47 53 eqtrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) = 𝑆 )
55 44 54 oveq12d ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) ∧ ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) ) = ( 𝑃 ∧ 𝑆 ) )
56 1 2 3 4 7 8 dalempnes ⊢ ( 𝜑 → 𝑃 ≠ 𝑆 )
57 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
58 12 57 syl ⊢ ( 𝜑 → 𝐾 ∈ AtLat )
59 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
60 6 59 4 atnem0 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑆 ↔ ( 𝑃 ∧ 𝑆 ) = ( 0. ‘ 𝐾 ) ) )
61 58 19 26 60 syl3anc ⊢ ( 𝜑 → ( 𝑃 ≠ 𝑆 ↔ ( 𝑃 ∧ 𝑆 ) = ( 0. ‘ 𝐾 ) ) )
62 56 61 mpbid ⊢ ( 𝜑 → ( 𝑃 ∧ 𝑆 ) = ( 0. ‘ 𝐾 ) )
63 62 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑃 ∧ 𝑆 ) = ( 0. ‘ 𝐾 ) )
64 34 55 63 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) )
65 58 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ AtLat )
66 1 2 3 4 5 6 7 8 9 10 dalem23 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ∈ 𝐴 )
67 21 2 6 59 4 atnle ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝐺 ∈ 𝐴 ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) → ( ¬ 𝐺 ≤ 𝑌 ↔ ( 𝐺 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) )
68 65 66 31 67 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ¬ 𝐺 ≤ 𝑌 ↔ ( 𝐺 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) )
69 64 68 mpbird ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝐺 ≤ 𝑌 )