Metamath Proof Explorer


Theorem dalemply

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012)

Ref Expression
Hypotheses dalema.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 ) ) ) ) )
dalemc.l = ( le ‘ 𝐾 )
dalemc.j = ( join ‘ 𝐾 )
dalemc.a 𝐴 = ( Atoms ‘ 𝐾 )
dalempnes.o 𝑂 = ( LPlanes ‘ 𝐾 )
dalempnes.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
Assertion dalemply ( 𝜑𝑃 𝑌 )

Proof

Step Hyp Ref Expression
1 dalema.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 dalemc.l = ( le ‘ 𝐾 )
3 dalemc.j = ( join ‘ 𝐾 )
4 dalemc.a 𝐴 = ( Atoms ‘ 𝐾 )
5 dalempnes.o 𝑂 = ( LPlanes ‘ 𝐾 )
6 dalempnes.y 𝑌 = ( ( 𝑃 𝑄 ) 𝑅 )
7 1 dalemkelat ( 𝜑𝐾 ∈ Lat )
8 1 4 dalempeb ( 𝜑𝑃 ∈ ( Base ‘ 𝐾 ) )
9 1 dalemkehl ( 𝜑𝐾 ∈ HL )
10 1 dalemqea ( 𝜑𝑄𝐴 )
11 1 dalemrea ( 𝜑𝑅𝐴 )
12 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
13 12 3 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴 ) → ( 𝑄 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
14 9 10 11 13 syl3anc ( 𝜑 → ( 𝑄 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
15 12 2 3 latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) → 𝑃 ( 𝑃 ( 𝑄 𝑅 ) ) )
16 7 8 14 15 syl3anc ( 𝜑𝑃 ( 𝑃 ( 𝑄 𝑅 ) ) )
17 1 dalempea ( 𝜑𝑃𝐴 )
18 3 4 hlatjass ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → ( ( 𝑃 𝑄 ) 𝑅 ) = ( 𝑃 ( 𝑄 𝑅 ) ) )
19 9 17 10 11 18 syl13anc ( 𝜑 → ( ( 𝑃 𝑄 ) 𝑅 ) = ( 𝑃 ( 𝑄 𝑅 ) ) )
20 16 19 breqtrrd ( 𝜑𝑃 ( ( 𝑃 𝑄 ) 𝑅 ) )
21 20 6 breqtrrdi ( 𝜑𝑃 𝑌 )