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 ) ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑆𝐴𝑇𝐴𝑈𝐴 ) ) ∧ ( 𝑌𝑂𝑍𝑂 ) ∧ ( ( ¬ 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 ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 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 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝐷 𝐸 ) = 𝐵 )

Proof

Step Hyp Ref Expression
1 dalem.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑆𝐴𝑇𝐴𝑈𝐴 ) ) ∧ ( 𝑌𝑂𝑍𝑂 ) ∧ ( ( ¬ 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 ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 𝑌 ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 ( 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 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐷 𝐵 )
17 1 2 3 4 5 6 7 8 9 11 12 13 14 15 dalem58 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐸 𝐵 )
18 1 dalemkelat ( 𝜑𝐾 ∈ Lat )
19 18 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐾 ∈ Lat )
20 1 2 3 4 6 7 8 9 10 dalemdea ( 𝜑𝐷𝐴 )
21 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
22 21 4 atbase ( 𝐷𝐴𝐷 ∈ ( Base ‘ 𝐾 ) )
23 20 22 syl ( 𝜑𝐷 ∈ ( Base ‘ 𝐾 ) )
24 23 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐷 ∈ ( Base ‘ 𝐾 ) )
25 1 2 3 4 6 7 8 9 11 dalemeea ( 𝜑𝐸𝐴 )
26 21 4 atbase ( 𝐸𝐴𝐸 ∈ ( Base ‘ 𝐾 ) )
27 25 26 syl ( 𝜑𝐸 ∈ ( Base ‘ 𝐾 ) )
28 27 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 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 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐵 ∈ ( LLines ‘ 𝐾 ) )
31 21 29 llnbase ( 𝐵 ∈ ( LLines ‘ 𝐾 ) → 𝐵 ∈ ( Base ‘ 𝐾 ) )
32 30 31 syl ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐵 ∈ ( Base ‘ 𝐾 ) )
33 21 2 3 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝐷 ∈ ( Base ‘ 𝐾 ) ∧ 𝐸 ∈ ( Base ‘ 𝐾 ) ∧ 𝐵 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐷 𝐵𝐸 𝐵 ) ↔ ( 𝐷 𝐸 ) 𝐵 ) )
34 19 24 28 32 33 syl13anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝐷 𝐵𝐸 𝐵 ) ↔ ( 𝐷 𝐸 ) 𝐵 ) )
35 16 17 34 mpbi2and ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝐷 𝐸 ) 𝐵 )
36 1 dalemkehl ( 𝜑𝐾 ∈ HL )
37 36 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐾 ∈ HL )
38 1 2 3 4 6 7 8 9 10 11 dalemdnee ( 𝜑𝐷𝐸 )
39 3 4 29 llni2 ( ( ( 𝐾 ∈ HL ∧ 𝐷𝐴𝐸𝐴 ) ∧ 𝐷𝐸 ) → ( 𝐷 𝐸 ) ∈ ( LLines ‘ 𝐾 ) )
40 36 20 25 38 39 syl31anc ( 𝜑 → ( 𝐷 𝐸 ) ∈ ( LLines ‘ 𝐾 ) )
41 40 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 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 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝐷 𝐸 ) 𝐵 ↔ ( 𝐷 𝐸 ) = 𝐵 ) )
44 35 43 mpbid ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝐷 𝐸 ) = 𝐵 )