Metamath Proof Explorer


Theorem cdleme20y

Description: Part of proof of Lemma E in Crawley p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012) (Proof shortened by OpenAI, 25-Mar-2020)

Ref Expression
Hypotheses cdleme20z.l = ( le ‘ 𝐾 )
cdleme20z.j = ( join ‘ 𝐾 )
cdleme20z.m = ( meet ‘ 𝐾 )
cdleme20z.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cdleme20y ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑆𝐴𝑇𝐴 ) ∧ ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑅 ) ( 𝑇 𝑅 ) ) = 𝑅 )

Proof

Step Hyp Ref Expression
1 cdleme20z.l = ( le ‘ 𝐾 )
2 cdleme20z.j = ( join ‘ 𝐾 )
3 cdleme20z.m = ( meet ‘ 𝐾 )
4 cdleme20z.a 𝐴 = ( Atoms ‘ 𝐾 )
5 simp1 ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑆𝐴𝑇𝐴 ) ∧ ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) ) → 𝐾 ∈ HL )
6 simp22 ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑆𝐴𝑇𝐴 ) ∧ ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) ) → 𝑆𝐴 )
7 simp23 ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑆𝐴𝑇𝐴 ) ∧ ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) ) → 𝑇𝐴 )
8 simp21 ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑆𝐴𝑇𝐴 ) ∧ ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) ) → 𝑅𝐴 )
9 simp3 ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑆𝐴𝑇𝐴 ) ∧ ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) ) → ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) )
10 1 2 3 4 2llnma2rN ( ( 𝐾 ∈ HL ∧ ( 𝑆𝐴𝑇𝐴𝑅𝐴 ) ∧ ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑅 ) ( 𝑇 𝑅 ) ) = 𝑅 )
11 5 6 7 8 9 10 syl131anc ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑆𝐴𝑇𝐴 ) ∧ ( 𝑆𝑇 ∧ ¬ 𝑅 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑅 ) ( 𝑇 𝑅 ) ) = 𝑅 )