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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 ) ) ∧ ( 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 dalem27 ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) )

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 1 dalemkelat ⊢ ( 𝜑 → 𝐾 ∈ Lat )
12 11 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ Lat )
13 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
14 13 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL )
15 5 dalemccea ⊢ ( 𝜓 → 𝑐 ∈ 𝐴 )
16 15 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ 𝐴 )
17 1 dalempea ⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
18 17 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ∈ 𝐴 )
19 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
20 19 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
21 14 16 18 20 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
22 5 dalemddea ⊢ ( 𝜓 → 𝑑 ∈ 𝐴 )
23 22 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑑 ∈ 𝐴 )
24 1 dalemsea ⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
25 24 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑆 ∈ 𝐴 )
26 19 3 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
27 14 23 25 26 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 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 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ≤ ( 𝑐 ∨ 𝑃 ) )
30 10 29 eqbrtrid ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ≤ ( 𝑐 ∨ 𝑃 ) )
31 1 2 3 4 5 6 7 8 9 10 dalem23 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ∈ 𝐴 )
32 1 2 3 4 7 8 dalemply ⊢ ( 𝜑 → 𝑃 ≤ 𝑌 )
33 32 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ≤ 𝑌 )
34 1 2 3 4 5 6 7 8 9 10 dalem24 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝐺 ≤ 𝑌 )
35 nbrne2 ⊢ ( ( 𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌 ) → 𝑃 ≠ 𝐺 )
36 35 necomd ⊢ ( ( 𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌 ) → 𝐺 ≠ 𝑃 )
37 33 34 36 syl2anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ≠ 𝑃 )
38 2 3 4 hlatexch2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝐺 ≠ 𝑃 ) → ( 𝐺 ≤ ( 𝑐 ∨ 𝑃 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ) )
39 14 31 16 18 37 38 syl131anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ≤ ( 𝑐 ∨ 𝑃 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ) )
40 30 39 mpd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) )