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Metamath Proof Explorer

Theorem cdleme43aN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v_1). (Contributed by NM, 20-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme43.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme43.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme43.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme43.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme43.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme43.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme43.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme43.x ⊢ 𝑋 = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑊 )
cdleme43.c ⊢ 𝐶 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme43.f ⊢ 𝑍 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐶 ∨ ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme43.d ⊢ 𝐷 = ( ( 𝑆 ∨ 𝑋 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme43.g ⊢ 𝐺 = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme43.e ⊢ 𝐸 = ( ( 𝐷 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝐷 ) ∧ 𝑊 ) ) )
cdleme43.v ⊢ 𝑉 = ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 )
cdleme43.y ⊢ 𝑌 = ( ( 𝑅 ∨ 𝐷 ) ∧ 𝑊 )
Assertion cdleme43aN ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ 𝑉 ) ) )

Proof

Step Hyp Ref Expression
1 cdleme43.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme43.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdleme43.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdleme43.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdleme43.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme43.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme43.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme43.x ⊢ 𝑋 = ( ( 𝑄 ∨ 𝑃 ) ∧ 𝑊 )
9 cdleme43.c ⊢ 𝐶 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
10 cdleme43.f ⊢ 𝑍 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐶 ∨ ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 ) ) )
11 cdleme43.d ⊢ 𝐷 = ( ( 𝑆 ∨ 𝑋 ) ∧ ( 𝑃 ∨ ( ( 𝑄 ∨ 𝑆 ) ∧ 𝑊 ) ) )
12 cdleme43.g ⊢ 𝐺 = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) ) )
13 cdleme43.e ⊢ 𝐸 = ( ( 𝐷 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝐷 ) ∧ 𝑊 ) ) )
14 cdleme43.v ⊢ 𝑉 = ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 )
15 cdleme43.y ⊢ 𝑌 = ( ( 𝑅 ∨ 𝐷 ) ∧ 𝑊 )
16 3 5 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) )
17 14 oveq2i ⊢ ( 𝐷 ∨ 𝑉 ) = ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) )
18 17 a1i ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐷 ∨ 𝑉 ) = ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) ) )
19 16 18 oveq12d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ 𝑉 ) ) = ( ( 𝑄 ∨ 𝑃 ) ∧ ( 𝐷 ∨ ( ( 𝑍 ∨ 𝑆 ) ∧ 𝑊 ) ) ) )
20 12 19 eqtr4id ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ 𝑉 ) ) )