Metamath Proof Explorer

Theorem cdlema2N

Description: A condition for required for proof of Lemma A in Crawley p. 112. (Contributed by NM, 9-May-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdlema2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlema2.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlema2.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlema2.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlema2.z ⊢ 0 = ( 0. ‘ 𝐾 )
cdlema2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion cdlema2N ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → ( 𝑅 ∧ 𝑋 ) = 0 )

Proof

Step Hyp Ref Expression
1 cdlema2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlema2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlema2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlema2.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlema2.z ⊢ 0 = ( 0. ‘ 𝐾 )
6 cdlema2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
7 simp3ll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → 𝑅 ≠ 𝑃 )
8 simp3rl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → 𝑃 ≤ 𝑋 )
9 simp3rr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → ¬ 𝑄 ≤ 𝑋 )
10 simp3lr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
11 8 9 10 3jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
12 1 2 3 6 exatleN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ≤ 𝑋 ↔ 𝑅 = 𝑃 ) )
13 11 12 syld3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → ( 𝑅 ≤ 𝑋 ↔ 𝑅 = 𝑃 ) )
14 13 necon3bbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → ( ¬ 𝑅 ≤ 𝑋 ↔ 𝑅 ≠ 𝑃 ) )
15 7 14 mpbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → ¬ 𝑅 ≤ 𝑋 )
16 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → 𝐾 ∈ HL )
17 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
18 16 17 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → 𝐾 ∈ AtLat )
19 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → 𝑅 ∈ 𝐴 )
20 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → 𝑋 ∈ 𝐵 )
21 1 2 4 5 6 atnle ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( ¬ 𝑅 ≤ 𝑋 ↔ ( 𝑅 ∧ 𝑋 ) = 0 ) )
22 18 19 20 21 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → ( ¬ 𝑅 ≤ 𝑋 ↔ ( 𝑅 ∧ 𝑋 ) = 0 ) )
23 15 22 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑅 ≠ 𝑃 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ) ) ) → ( 𝑅 ∧ 𝑋 ) = 0 )