Metamath Proof Explorer


Theorem dalem27

Description: Lemma for dath . Show that the line G P intersects the dummy center of perspectivity c . (Contributed by NM, 8-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 ) ) ) )
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 dalem27 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐 ( 𝐺 𝑃 ) )

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 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 1 dalemkelat ( 𝜑𝐾 ∈ Lat )
12 11 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐾 ∈ Lat )
13 1 dalemkehl ( 𝜑𝐾 ∈ HL )
14 13 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐾 ∈ HL )
15 5 dalemccea ( 𝜓𝑐𝐴 )
16 15 3ad2ant3 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐𝐴 )
17 1 dalempea ( 𝜑𝑃𝐴 )
18 17 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑃𝐴 )
19 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
20 19 3 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑐𝐴𝑃𝐴 ) → ( 𝑐 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
21 14 16 18 20 syl3anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝑐 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
22 5 dalemddea ( 𝜓𝑑𝐴 )
23 22 3ad2ant3 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑑𝐴 )
24 1 dalemsea ( 𝜑𝑆𝐴 )
25 24 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑆𝐴 )
26 19 3 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑑𝐴𝑆𝐴 ) → ( 𝑑 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
27 14 23 25 26 syl3anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝑑 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
28 19 2 6 latmle1 ( ( 𝐾 ∈ Lat ∧ ( 𝑐 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑑 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑐 𝑃 ) ( 𝑑 𝑆 ) ) ( 𝑐 𝑃 ) )
29 12 21 27 28 syl3anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝑐 𝑃 ) ( 𝑑 𝑆 ) ) ( 𝑐 𝑃 ) )
30 10 29 eqbrtrid ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐺 ( 𝑐 𝑃 ) )
31 1 2 3 4 5 6 7 8 9 10 dalem23 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐺𝐴 )
32 1 2 3 4 7 8 dalemply ( 𝜑𝑃 𝑌 )
33 32 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑃 𝑌 )
34 1 2 3 4 5 6 7 8 9 10 dalem24 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ¬ 𝐺 𝑌 )
35 nbrne2 ( ( 𝑃 𝑌 ∧ ¬ 𝐺 𝑌 ) → 𝑃𝐺 )
36 35 necomd ( ( 𝑃 𝑌 ∧ ¬ 𝐺 𝑌 ) → 𝐺𝑃 )
37 33 34 36 syl2anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐺𝑃 )
38 2 3 4 hlatexch2 ( ( 𝐾 ∈ HL ∧ ( 𝐺𝐴𝑐𝐴𝑃𝐴 ) ∧ 𝐺𝑃 ) → ( 𝐺 ( 𝑐 𝑃 ) → 𝑐 ( 𝐺 𝑃 ) ) )
39 14 31 16 18 37 38 syl131anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝐺 ( 𝑐 𝑃 ) → 𝑐 ( 𝐺 𝑃 ) ) )
40 30 39 mpd ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐 ( 𝐺 𝑃 ) )