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Metamath Proof Explorer

Theorem cdleme10tN

Description: Part of proof of Lemma E in Crawley p. 113, 2nd paragraph on p. 114. Y represents t_2. In their notation, we prove t \/ t_2 = t \/ r. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme10t.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme10t.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme10t.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme10t.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme10t.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme10t.y ⊢ 𝑌 = ( ( 𝑅 ∨ 𝑇 ) ∧ 𝑊 )
Assertion cdleme10tN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑅 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) → ( 𝑇 ∨ 𝑌 ) = ( 𝑇 ∨ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 cdleme10t.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme10t.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme10t.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme10t.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme10t.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme10t.y ⊢ 𝑌 = ( ( 𝑅 ∨ 𝑇 ) ∧ 𝑊 )
7 1 2 3 4 5 6 cdleme10 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑅 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) → ( 𝑇 ∨ 𝑌 ) = ( 𝑇 ∨ 𝑅 ) )