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Metamath Proof Explorer

Theorem cdleme17a

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. F , G , and C represent f(s), f_s(p), and s_1 respectively. We show, in their notation, f_s(p)=(p \/ q) /\ (q \/ s_1). (Contributed by NM, 11-Oct-2012)

Ref Expression
Hypotheses cdleme17.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme17.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme17.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme17.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme17.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme17.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme17.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme17.g ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme17.c ⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
Assertion cdleme17a ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 cdleme17.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme17.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme17.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme17.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme17.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme17.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme17.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8 cdleme17.g ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
9 cdleme17.c ⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
10 1 2 3 4 5 6 7 8 9 cdleme7a ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐶 ) )
11 1 2 3 4 5 6 7 9 cdleme9 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ∨ 𝐶 ) = ( 𝑄 ∨ 𝐶 ) )
12 11 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐶 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝐶 ) ) )
13 10 12 eqtrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑄 ∨ 𝐶 ) ) )