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Metamath Proof Explorer

Theorem cdleme9b

Description: Utility lemma for Lemma E in Crawley p. 113. (Contributed by NM, 9-Oct-2012)

Ref Expression
Hypotheses cdleme9b.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme9b.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme9b.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme9b.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme9b.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme9b.c ⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
Assertion cdleme9b ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) → 𝐶 ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 cdleme9b.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme9b.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme9b.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme9b.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme9b.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme9b.c ⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
7 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
8 7 adantr ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) → 𝐾 ∈ Lat )
9 1 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑆 ) ∈ 𝐵 )
10 9 3adant3r3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) → ( 𝑃 ∨ 𝑆 ) ∈ 𝐵 )
11 simpr3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) → 𝑊 ∈ 𝐻 )
12 1 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
13 11 12 syl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) → 𝑊 ∈ 𝐵 )
14 1 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑆 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ 𝐵 )
15 8 10 13 14 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ 𝐵 )
16 6 15 eqeltrid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) → 𝐶 ∈ 𝐵 )