Metamath Proof Explorer


Theorem dalem28

Description: Lemma for dath . Lemma dalem27 expressed differently. (Contributed by NM, 4-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 dalem28 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑃 ( 𝐺 𝑐 ) )

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 2 3 4 5 6 7 8 9 10 dalem27 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐 ( 𝐺 𝑃 ) )
12 1 dalemkehl ( 𝜑𝐾 ∈ HL )
13 12 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐾 ∈ HL )
14 5 dalemccea ( 𝜓𝑐𝐴 )
15 14 3ad2ant3 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐𝐴 )
16 1 dalempea ( 𝜑𝑃𝐴 )
17 16 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑃𝐴 )
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 10 dalem25 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑐𝐺 )
20 2 3 4 hlatexch1 ( ( 𝐾 ∈ HL ∧ ( 𝑐𝐴𝑃𝐴𝐺𝐴 ) ∧ 𝑐𝐺 ) → ( 𝑐 ( 𝐺 𝑃 ) → 𝑃 ( 𝐺 𝑐 ) ) )
21 13 15 17 18 19 20 syl131anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝑐 ( 𝐺 𝑃 ) → 𝑃 ( 𝐺 𝑐 ) ) )
22 11 21 mpd ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝑃 ( 𝐺 𝑐 ) )