Metamath Proof Explorer


Theorem dalem30

Description: Lemma for dath . Analogue of dalem24 for H . (Contributed by NM, 3-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 ) ) ) )
dalem29.m = ( meet ‘ 𝐾 )
dalem29.o 𝑂 = ( LPlanes ‘ 𝐾 )
dalem29.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
dalem29.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
dalem29.h 𝐻 = ( ( 𝑐 𝑄 ) ( 𝑑 𝑇 ) )
Assertion dalem30 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ¬ 𝐻 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 dalem29.m = ( meet ‘ 𝐾 )
7 dalem29.o 𝑂 = ( LPlanes ‘ 𝐾 )
8 dalem29.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
9 dalem29.z 𝑍 = ( ( 𝑆 𝑇 ) 𝑈 )
10 dalem29.h 𝐻 = ( ( 𝑐 𝑄 ) ( 𝑑 𝑇 ) )
11 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 ) ) ) ) )
12 11 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 ) ) ) ) )
13 1 2 3 4 8 9 dalemrotyz ( ( 𝜑𝑌 = 𝑍 ) → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑇 𝑈 ) 𝑆 ) )
14 13 3adant3 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑇 𝑈 ) 𝑆 ) )
15 1 2 3 4 5 8 dalemrotps ( ( 𝜑𝜓 ) → ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ 𝐶 ( 𝑐 𝑑 ) ) ) )
16 15 3adant2 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( ( 𝑐𝐴𝑑𝐴 ) ∧ ¬ 𝑐 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ ( 𝑑𝑐 ∧ ¬ 𝑑 ( ( 𝑄 𝑅 ) 𝑃 ) ∧ 𝐶 ( 𝑐 𝑑 ) ) ) )
17 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 ) ) ) ) )
18 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 ) ) ) )
19 eqid ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑄 𝑅 ) 𝑃 )
20 eqid ( ( 𝑇 𝑈 ) 𝑆 ) = ( ( 𝑇 𝑈 ) 𝑆 )
21 17 2 3 4 18 6 7 19 20 10 dalem24 ( ( ( ( ( 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 ) ) ) ) → ¬ 𝐻 ( ( 𝑄 𝑅 ) 𝑃 ) )
22 12 14 16 21 syl3anc ( ( 𝜑𝑌 = 𝑍𝜓 ) → ¬ 𝐻 ( ( 𝑄 𝑅 ) 𝑃 ) )
23 1 3 4 dalemqrprot ( 𝜑 → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑃 𝑄 ) 𝑅 ) )
24 8 23 eqtr4id ( 𝜑𝑌 = ( ( 𝑄 𝑅 ) 𝑃 ) )
25 24 breq2d ( 𝜑 → ( 𝐻 𝑌𝐻 ( ( 𝑄 𝑅 ) 𝑃 ) ) )
26 25 3ad2ant1 ( ( 𝜑𝑌 = 𝑍𝜓 ) → ( 𝐻 𝑌𝐻 ( ( 𝑄 𝑅 ) 𝑃 ) ) )
27 22 26 mtbird ( ( 𝜑𝑌 = 𝑍𝜓 ) → ¬ 𝐻 𝑌 )