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Metamath Proof Explorer

Theorem dalemswapyzps

Description: Lemma for dath . Swap the Y and Z planes, along with dummy concurrency (center of perspectivity) atoms c and d , to allow reuse of analogous proofs. (Contributed by NM, 17-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 ) ) ) )
Assertion dalemswapyzps ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑑 ≤ 𝑍 ∧ ( 𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ ( 𝑑 ∨ 𝑐 ) ) ) )

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 5 dalemddea ⊢ ( 𝜓 → 𝑑 ∈ 𝐴 )
7 5 dalemccea ⊢ ( 𝜓 → 𝑐 ∈ 𝐴 )
8 6 7 jca ⊢ ( 𝜓 → ( 𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ) )
9 8 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ) )
10 5 dalem-ddly ⊢ ( 𝜓 → ¬ 𝑑 ≤ 𝑌 )
11 10 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑑 ≤ 𝑌 )
12 simp2 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑌 = 𝑍 )
13 12 breq2d ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ≤ 𝑌 ↔ 𝑑 ≤ 𝑍 ) )
14 11 13 mtbid ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑑 ≤ 𝑍 )
15 5 dalemccnedd ⊢ ( 𝜓 → 𝑐 ≠ 𝑑 )
16 15 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≠ 𝑑 )
17 5 dalem-ccly ⊢ ( 𝜓 → ¬ 𝑐 ≤ 𝑌 )
18 17 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ 𝑌 )
19 12 breq2d ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ≤ 𝑌 ↔ 𝑐 ≤ 𝑍 ) )
20 18 19 mtbid ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ 𝑍 )
21 5 dalemclccjdd ⊢ ( 𝜓 → 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) )
22 21 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) )
23 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
24 23 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL )
25 7 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ 𝐴 )
26 6 3ad2ant3 ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑑 ∈ 𝐴 )
27 3 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑑 ) = ( 𝑑 ∨ 𝑐 ) )
28 24 25 26 27 syl3anc ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑑 ) = ( 𝑑 ∨ 𝑐 ) )
29 22 28 breqtrd ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐶 ≤ ( 𝑑 ∨ 𝑐 ) )
30 16 20 29 3jca ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ ( 𝑑 ∨ 𝑐 ) ) )
31 9 14 30 3jca ⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑑 ≤ 𝑍 ∧ ( 𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ ( 𝑑 ∨ 𝑐 ) ) ) )