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Metamath Proof Explorer

Theorem cdleme42d

Description: Part of proof of Lemma E in Crawley p. 113. Match ( s .\/ ( x ./\ W ) ) = x . (Contributed by NM, 6-Mar-2013)

Ref Expression
Hypotheses cdleme42.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme42.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme42.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme42.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme42.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme42.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme42.v ⊢ 𝑉 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 )
Assertion cdleme42d ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) → ( 𝑅 ∨ ( ( 𝑅 ∨ 𝑉 ) ∧ 𝑊 ) ) = ( 𝑅 ∨ 𝑉 ) )

Proof

Step Hyp Ref Expression
1 cdleme42.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme42.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdleme42.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdleme42.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdleme42.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme42.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme42.v ⊢ 𝑉 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 )
8 7 oveq2i ⊢ ( 𝑅 ∨ 𝑉 ) = ( 𝑅 ∨ ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 ) )
9 1 2 3 4 5 6 7 cdleme42a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) → ( 𝑅 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑉 ) )
10 9 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) → ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 ) = ( ( 𝑅 ∨ 𝑉 ) ∧ 𝑊 ) )
11 10 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) → ( 𝑅 ∨ ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 ) ) = ( 𝑅 ∨ ( ( 𝑅 ∨ 𝑉 ) ∧ 𝑊 ) ) )
12 8 11 eqtr2id ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) → ( 𝑅 ∨ ( ( 𝑅 ∨ 𝑉 ) ∧ 𝑊 ) ) = ( 𝑅 ∨ 𝑉 ) )