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Metamath Proof Explorer

Theorem cdlemeulpq

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 5-Dec-2012)

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 cdlemeulpq ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 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 simpll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ HL )
8 7 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ Lat )
9 simprl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 )
10 simprr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 )
11 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12 11 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
13 7 9 10 12 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
14 11 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
15 14 ad2antlr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
16 11 1 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
17 8 13 15 16 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
18 6 17 eqbrtrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) )