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Metamath Proof Explorer

Theorem dalem57

Description: Lemma for dath . Axis of perspectivity point D is on the auxiliary line B . (Contributed by NM, 9-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 ) ) ) )
dalem57.m ⊢ ∧ = ( meet ‘ 𝐾 )
dalem57.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem57.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem57.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem57.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
dalem57.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
dalem57.h ⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) )
dalem57.i ⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) )
dalem57.b1 ⊢ 𝐵 = ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∧ 𝑌 )
Assertion dalem57 ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐷 ≤ 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 dalem57.m ⊢ ∧ = ( meet ‘ 𝐾 )
7 dalem57.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem57.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
9 dalem57.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
10 dalem57.d ⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) )
11 dalem57.g ⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) )
12 dalem57.h ⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) )
13 dalem57.i ⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) )
14 dalem57.b1 ⊢ 𝐵 = ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∧ 𝑌 )
15 1 2 3 4 5 6 7 8 9 11 12 13 14 dalem55 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) )
16 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
17 16 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ Lat )
18 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
19 18 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL )
20 1 2 3 4 5 6 7 8 9 11 dalem23 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ∈ 𝐴 )
21 1 2 3 4 5 6 7 8 9 12 dalem29 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐻 ∈ 𝐴 )
22 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
23 22 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ) → ( 𝐺 ∨ 𝐻 ) ∈ ( Base ‘ 𝐾 ) )
24 19 20 21 23 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ∨ 𝐻 ) ∈ ( Base ‘ 𝐾 ) )
25 1 3 4 dalempjqeb ⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
26 25 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
27 22 2 6 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐺 ∨ 𝐻 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )
28 17 24 26 27 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑃 ∨ 𝑄 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )
29 15 28 eqbrtrrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( 𝑃 ∨ 𝑄 ) )
30 1 2 3 4 5 6 7 8 9 11 12 13 14 dalem56 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑆 ∨ 𝑇 ) ) = ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) )
31 1 3 4 dalemsjteb ⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
32 31 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
33 22 2 6 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐺 ∨ 𝐻 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( 𝑆 ∨ 𝑇 ) )
34 17 24 32 33 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( 𝑆 ∨ 𝑇 ) )
35 30 34 eqbrtrrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( 𝑆 ∨ 𝑇 ) )
36 1 2 3 4 5 6 7 8 9 11 12 13 14 dalem54 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ∈ 𝐴 )
37 22 4 atbase ⊢ ( ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ∈ 𝐴 → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ∈ ( Base ‘ 𝐾 ) )
38 36 37 syl ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ∈ ( Base ‘ 𝐾 ) )
39 22 2 6 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( 𝑆 ∨ 𝑇 ) ) ↔ ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ) )
40 17 38 26 32 39 syl13anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( 𝑆 ∨ 𝑇 ) ) ↔ ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ) )
41 29 35 40 mpbi2and ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) )
42 41 10 breqtrrdi ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ 𝐷 )
43 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
44 19 43 syl ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ AtLat )
45 1 2 3 4 6 7 8 9 10 dalemdea ⊢ ( 𝜑 → 𝐷 ∈ 𝐴 )
46 45 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐷 ∈ 𝐴 )
47 2 4 atcmp ⊢ ( ( 𝐾 ∈ AtLat ∧ ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ 𝐷 ↔ ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) = 𝐷 ) )
48 44 36 46 47 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ 𝐷 ↔ ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) = 𝐷 ) )
49 42 48 mpbid ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) = 𝐷 )
50 eqid ⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 )
51 1 2 3 4 5 6 50 7 8 9 11 12 13 14 dalem53 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐵 ∈ ( LLines ‘ 𝐾 ) )
52 22 50 llnbase ⊢ ( 𝐵 ∈ ( LLines ‘ 𝐾 ) → 𝐵 ∈ ( Base ‘ 𝐾 ) )
53 51 52 syl ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐵 ∈ ( Base ‘ 𝐾 ) )
54 22 2 6 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐺 ∨ 𝐻 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝐵 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ 𝐵 )
55 17 24 53 54 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∧ 𝐵 ) ≤ 𝐵 )
56 49 55 eqbrtrrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐷 ≤ 𝐵 )