Metamath Proof Explorer

Theorem cdleme13

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, " and are centrally perspective". F and G represent f(s) and f(t) respectively. (Contributed by NM, 7-Oct-2012)

Ref Expression
Hypotheses cdleme12.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme12.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme12.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme12.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme12.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme12.g ⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
Assertion cdleme13 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )

Proof

Step Hyp Ref Expression
1 cdleme12.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme12.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme12.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme12.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme12.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8 cdleme12.g ⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
9 1 2 3 4 5 6 7 8 cdleme12 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) = 𝑈 )
10 9 6 eqtrdi ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) )
11 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝐾 ∈ HL )
12 11 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝐾 ∈ Lat )
13 simp21l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑃 ∈ 𝐴 )
14 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑄 ∈ 𝐴 )
15 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
16 15 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
17 11 13 14 16 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
18 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑊 ∈ 𝐻 )
19 15 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
20 18 19 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
21 15 1 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
22 12 17 20 21 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
23 10 22 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ≠ 𝑇 ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )