Metamath Proof Explorer

Theorem cdleme0cq

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 25-Apr-2013)

Ref Expression
Hypotheses cdleme0.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme0.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme0.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
Assertion cdleme0cq ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑄 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) )

Proof

Step Hyp Ref Expression
1 cdleme0.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme0.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme0.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme0.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme0.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 6 oveq2i ⊢ ( 𝑄 ∨ 𝑈 ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) )
8 simpll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝐾 ∈ HL )
9 simprrl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝑄 ∈ 𝐴 )
10 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
11 10 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝐾 ∈ Lat )
12 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
13 12 4 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
14 13 ad2antrl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
15 12 4 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
16 9 15 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
17 12 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
18 11 14 16 17 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
19 12 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
20 19 ad2antlr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
21 12 1 2 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) )
22 11 14 16 21 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) )
23 12 1 2 3 4 atmod3i1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝑊 ) ) )
24 8 9 18 20 22 23 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝑊 ) ) )
25 eqid ⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
26 1 2 25 4 5 lhpjat2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑄 ∨ 𝑊 ) = ( 1. ‘ 𝐾 ) )
27 26 adantrl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑄 ∨ 𝑊 ) = ( 1. ‘ 𝐾 ) )
28 27 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝑊 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 1. ‘ 𝐾 ) ) )
29 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
30 29 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → 𝐾 ∈ OL )
31 12 3 25 olm11 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ 𝑄 ) )
32 30 18 31 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 1. ‘ 𝐾 ) ) = ( 𝑃 ∨ 𝑄 ) )
33 24 28 32 3eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) = ( 𝑃 ∨ 𝑄 ) )
34 7 33 eqtrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝑄 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) )