Metamath Proof Explorer


Theorem dalem51

Description: Lemma for dath . Construct the condition ph with c , G H I , and Y in place of C , Y , and Z respectively. This lets us reuse the special case of Desargues's theorem where Y =/= Z , to eventually prove the case where Y = Z . (Contributed by NM, 16-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 ) ) ) )
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 dalem51 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑐𝐴 ) ∧ ( 𝐺𝐴𝐻𝐴𝐼𝐴 ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) ∧ ( ( ( 𝐺 𝐻 ) 𝐼 ) ∈ 𝑂𝑌𝑂 ) ∧ ( ( ¬ 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 ) 𝐼 ) ≠ 𝑌 ) )

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 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 ( 𝜑𝐾 ∈ HL )
14 13 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐾 ∈ HL )
15 5 dalemccea ( 𝜓𝑐𝐴 )
16 15 3ad2ant3 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐𝐴 )
17 14 16 jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝐾 ∈ HL ∧ 𝑐𝐴 ) )
18 1 2 3 4 5 6 7 8 9 10 dalem23 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐺𝐴 )
19 1 2 3 4 5 6 7 8 9 11 dalem29 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐻𝐴 )
20 1 2 3 4 5 6 7 8 9 12 dalem34 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐼𝐴 )
21 18 19 20 3jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝐺𝐴𝐻𝐴𝐼𝐴 ) )
22 1 dalempea ( 𝜑𝑃𝐴 )
23 1 dalemqea ( 𝜑𝑄𝐴 )
24 1 dalemrea ( 𝜑𝑅𝐴 )
25 22 23 24 3jca ( 𝜑 → ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) )
26 25 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) )
27 17 21 26 3jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝐾 ∈ HL ∧ 𝑐𝐴 ) ∧ ( 𝐺𝐴𝐻𝐴𝐼𝐴 ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) )
28 1 2 3 4 5 6 7 8 9 10 11 12 dalem42 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝐺 𝐻 ) 𝐼 ) ∈ 𝑂 )
29 1 dalemyeo ( 𝜑𝑌𝑂 )
30 29 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑌𝑂 )
31 28 30 jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( ( 𝐺 𝐻 ) 𝐼 ) ∈ 𝑂𝑌𝑂 ) )
32 1 2 3 4 5 6 7 8 9 10 11 12 dalem45 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ¬ 𝑐 ( 𝐺 𝐻 ) )
33 1 2 3 4 5 6 7 8 9 10 11 12 dalem46 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ¬ 𝑐 ( 𝐻 𝐼 ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 dalem47 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ¬ 𝑐 ( 𝐼 𝐺 ) )
35 32 33 34 3jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ¬ 𝑐 ( 𝐺 𝐻 ) ∧ ¬ 𝑐 ( 𝐻 𝐼 ) ∧ ¬ 𝑐 ( 𝐼 𝐺 ) ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 dalem48 ( ( 𝜑𝜓 ) → ¬ 𝑐 ( 𝑃 𝑄 ) )
37 1 2 3 4 5 6 7 8 9 10 11 12 dalem49 ( ( 𝜑𝜓 ) → ¬ 𝑐 ( 𝑄 𝑅 ) )
38 1 2 3 4 5 6 7 8 9 10 11 12 dalem50 ( ( 𝜑𝜓 ) → ¬ 𝑐 ( 𝑅 𝑃 ) )
39 36 37 38 3jca ( ( 𝜑𝜓 ) → ( ¬ 𝑐 ( 𝑃 𝑄 ) ∧ ¬ 𝑐 ( 𝑄 𝑅 ) ∧ ¬ 𝑐 ( 𝑅 𝑃 ) ) )
40 39 3adant2 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ¬ 𝑐 ( 𝑃 𝑄 ) ∧ ¬ 𝑐 ( 𝑄 𝑅 ) ∧ ¬ 𝑐 ( 𝑅 𝑃 ) ) )
41 1 2 3 4 5 6 7 8 9 10 dalem27 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐 ( 𝐺 𝑃 ) )
42 1 2 3 4 5 6 7 8 9 11 dalem32 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐 ( 𝐻 𝑄 ) )
43 1 2 3 4 5 6 7 8 9 12 dalem36 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐 ( 𝐼 𝑅 ) )
44 41 42 43 3jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝑐 ( 𝐺 𝑃 ) ∧ 𝑐 ( 𝐻 𝑄 ) ∧ 𝑐 ( 𝐼 𝑅 ) ) )
45 35 40 44 3jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( ¬ 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 ) ) ) )
46 27 31 45 3jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( ( 𝐾 ∈ HL ∧ 𝑐𝐴 ) ∧ ( 𝐺𝐴𝐻𝐴𝐼𝐴 ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) ∧ ( ( ( 𝐺 𝐻 ) 𝐼 ) ∈ 𝑂𝑌𝑂 ) ∧ ( ( ¬ 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 ) ) ) ) )
47 1 2 3 4 5 6 7 8 9 10 11 12 dalem43 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝐺 𝐻 ) 𝐼 ) ≠ 𝑌 )
48 46 47 jca ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑐𝐴 ) ∧ ( 𝐺𝐴𝐻𝐴𝐼𝐴 ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) ∧ ( ( ( 𝐺 𝐻 ) 𝐼 ) ∈ 𝑂𝑌𝑂 ) ∧ ( ( ¬ 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 ) 𝐼 ) ≠ 𝑌 ) )