Metamath Proof Explorer


Theorem dalem59

Description: Lemma for dath . Analogue of dalem57 for F . (Contributed by NM, 10-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 ) ) ) )
dalem59.m = ( meet ‘ 𝐾 )
dalem59.o 𝑂 = ( LPlanes ‘ 𝐾 )
dalem59.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
dalem59.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
dalem59.f 𝐹 = ( ( 𝑅 𝑃 ) ( 𝑈 𝑆 ) )
dalem59.g 𝐺 = ( ( 𝑐 𝑃 ) ( 𝑑 𝑆 ) )
dalem59.h 𝐻 = ( ( 𝑐 𝑄 ) ( 𝑑 𝑇 ) )
dalem59.i 𝐼 = ( ( 𝑐 𝑅 ) ( 𝑑 𝑈 ) )
dalem59.b1 𝐵 = ( ( ( 𝐺 𝐻 ) 𝐼 ) 𝑌 )
Assertion dalem59 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐹 𝐵 )

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 dalem59.m = ( meet ‘ 𝐾 )
7 dalem59.o 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem59.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
9 dalem59.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
10 dalem59.f 𝐹 = ( ( 𝑅 𝑃 ) ( 𝑈 𝑆 ) )
11 dalem59.g 𝐺 = ( ( 𝑐 𝑃 ) ( 𝑑 𝑆 ) )
12 dalem59.h 𝐻 = ( ( 𝑐 𝑄 ) ( 𝑑 𝑇 ) )
13 dalem59.i 𝐼 = ( ( 𝑐 𝑅 ) ( 𝑑 𝑈 ) )
14 dalem59.b1 𝐵 = ( ( ( 𝐺 𝐻 ) 𝐼 ) 𝑌 )
15 1 2 3 4 8 9 dalemrot ( 𝜑 → ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑄𝐴𝑅𝐴𝑃𝐴 ) ∧ ( 𝑇𝐴𝑈𝐴𝑆𝐴 ) ) ∧ ( ( ( 𝑄 𝑅 ) 𝑃 ) ∈ 𝑂 ∧ ( ( 𝑇 𝑈 ) 𝑆 ) ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ( 𝑄 𝑅 ) ∧ ¬ 𝐶 ( 𝑅 𝑃 ) ∧ ¬ 𝐶 ( 𝑃 𝑄 ) ) ∧ ( ¬ 𝐶 ( 𝑇 𝑈 ) ∧ ¬ 𝐶 ( 𝑈 𝑆 ) ∧ ¬ 𝐶 ( 𝑆 𝑇 ) ) ∧ ( 𝐶 ( 𝑄 𝑇 ) ∧ 𝐶 ( 𝑅 𝑈 ) ∧ 𝐶 ( 𝑃 𝑆 ) ) ) ) )
16 15 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑄𝐴𝑅𝐴𝑃𝐴 ) ∧ ( 𝑇𝐴𝑈𝐴𝑆𝐴 ) ) ∧ ( ( ( 𝑄 𝑅 ) 𝑃 ) ∈ 𝑂 ∧ ( ( 𝑇 𝑈 ) 𝑆 ) ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ( 𝑄 𝑅 ) ∧ ¬ 𝐶 ( 𝑅 𝑃 ) ∧ ¬ 𝐶 ( 𝑃 𝑄 ) ) ∧ ( ¬ 𝐶 ( 𝑇 𝑈 ) ∧ ¬ 𝐶 ( 𝑈 𝑆 ) ∧ ¬ 𝐶 ( 𝑆 𝑇 ) ) ∧ ( 𝐶 ( 𝑄 𝑇 ) ∧ 𝐶 ( 𝑅 𝑈 ) ∧ 𝐶 ( 𝑃 𝑆 ) ) ) ) )
17 1 2 3 4 8 9 dalemrotyz ( ( 𝜑𝑌 = 𝑍 ) → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑇 𝑈 ) 𝑆 ) )
18 17 3adant3 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑇 𝑈 ) 𝑆 ) )
19 1 2 3 4 5 8 dalemrotps ( ( 𝜑𝜓 ) → ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ 𝐶 ( 𝑐 𝑑 ) ) ) )
20 19 3adant2 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ 𝐶 ( 𝑐 𝑑 ) ) ) )
21 biid ( ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑄𝐴𝑅𝐴𝑃𝐴 ) ∧ ( 𝑇𝐴𝑈𝐴𝑆𝐴 ) ) ∧ ( ( ( 𝑄 𝑅 ) 𝑃 ) ∈ 𝑂 ∧ ( ( 𝑇 𝑈 ) 𝑆 ) ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ( 𝑄 𝑅 ) ∧ ¬ 𝐶 ( 𝑅 𝑃 ) ∧ ¬ 𝐶 ( 𝑃 𝑄 ) ) ∧ ( ¬ 𝐶 ( 𝑇 𝑈 ) ∧ ¬ 𝐶 ( 𝑈 𝑆 ) ∧ ¬ 𝐶 ( 𝑆 𝑇 ) ) ∧ ( 𝐶 ( 𝑄 𝑇 ) ∧ 𝐶 ( 𝑅 𝑈 ) ∧ 𝐶 ( 𝑃 𝑆 ) ) ) ) ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑄𝐴𝑅𝐴𝑃𝐴 ) ∧ ( 𝑇𝐴𝑈𝐴𝑆𝐴 ) ) ∧ ( ( ( 𝑄 𝑅 ) 𝑃 ) ∈ 𝑂 ∧ ( ( 𝑇 𝑈 ) 𝑆 ) ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ( 𝑄 𝑅 ) ∧ ¬ 𝐶 ( 𝑅 𝑃 ) ∧ ¬ 𝐶 ( 𝑃 𝑄 ) ) ∧ ( ¬ 𝐶 ( 𝑇 𝑈 ) ∧ ¬ 𝐶 ( 𝑈 𝑆 ) ∧ ¬ 𝐶 ( 𝑆 𝑇 ) ) ∧ ( 𝐶 ( 𝑄 𝑇 ) ∧ 𝐶 ( 𝑅 𝑈 ) ∧ 𝐶 ( 𝑃 𝑆 ) ) ) ) )
22 biid ( ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ 𝐶 ( 𝑐 𝑑 ) ) ) ↔ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ 𝐶 ( 𝑐 𝑑 ) ) ) )
23 eqid ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑄 𝑅 ) 𝑃 )
24 eqid ( ( 𝑇 𝑈 ) 𝑆 ) = ( ( 𝑇 𝑈 ) 𝑆 )
25 eqid ( ( ( 𝐻 𝐼 ) 𝐺 ) ( ( 𝑄 𝑅 ) 𝑃 ) ) = ( ( ( 𝐻 𝐼 ) 𝐺 ) ( ( 𝑄 𝑅 ) 𝑃 ) )
26 21 2 3 4 22 6 7 23 24 10 12 13 11 25 dalem58 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑄𝐴𝑅𝐴𝑃𝐴 ) ∧ ( 𝑇𝐴𝑈𝐴𝑆𝐴 ) ) ∧ ( ( ( 𝑄 𝑅 ) 𝑃 ) ∈ 𝑂 ∧ ( ( 𝑇 𝑈 ) 𝑆 ) ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ( 𝑄 𝑅 ) ∧ ¬ 𝐶 ( 𝑅 𝑃 ) ∧ ¬ 𝐶 ( 𝑃 𝑄 ) ) ∧ ( ¬ 𝐶 ( 𝑇 𝑈 ) ∧ ¬ 𝐶 ( 𝑈 𝑆 ) ∧ ¬ 𝐶 ( 𝑆 𝑇 ) ) ∧ ( 𝐶 ( 𝑄 𝑇 ) ∧ 𝐶 ( 𝑅 𝑈 ) ∧ 𝐶 ( 𝑃 𝑆 ) ) ) ) ∧ ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑇 𝑈 ) 𝑆 ) ∧ ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ 𝐶 ( 𝑐 𝑑 ) ) ) ) → 𝐹 ( ( ( 𝐻 𝐼 ) 𝐺 ) ( ( 𝑄 𝑅 ) 𝑃 ) ) )
27 16 18 20 26 syl3anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐹 ( ( ( 𝐻 𝐼 ) 𝐺 ) ( ( 𝑄 𝑅 ) 𝑃 ) ) )
28 1 dalemkehl ( 𝜑𝐾 ∈ HL )
29 28 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐾 ∈ HL )
30 1 2 3 4 5 6 7 8 9 12 dalem29 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐻𝐴 )
31 1 2 3 4 5 6 7 8 9 13 dalem34 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐼𝐴 )
32 1 2 3 4 5 6 7 8 9 11 dalem23 ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐺𝐴 )
33 3 4 hlatjrot ( ( 𝐾 ∈ HL ∧ ( 𝐻𝐴𝐼𝐴𝐺𝐴 ) ) → ( ( 𝐻 𝐼 ) 𝐺 ) = ( ( 𝐺 𝐻 ) 𝐼 ) )
34 29 30 31 32 33 syl13anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝐻 𝐼 ) 𝐺 ) = ( ( 𝐺 𝐻 ) 𝐼 ) )
35 1 3 4 dalemqrprot ( 𝜑 → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑃 𝑄 ) 𝑅 ) )
36 35 8 eqtr4di ( 𝜑 → ( ( 𝑄 𝑅 ) 𝑃 ) = 𝑌 )
37 36 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝑄 𝑅 ) 𝑃 ) = 𝑌 )
38 34 37 oveq12d ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( ( 𝐻 𝐼 ) 𝐺 ) ( ( 𝑄 𝑅 ) 𝑃 ) ) = ( ( ( 𝐺 𝐻 ) 𝐼 ) 𝑌 ) )
39 38 14 eqtr4di ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( ( 𝐻 𝐼 ) 𝐺 ) ( ( 𝑄 𝑅 ) 𝑃 ) ) = 𝐵 )
40 27 39 breqtrd ( ( 𝜑𝑌 = 𝑍𝜓 ) → 𝐹 𝐵 )