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 ∧ 𝑃𝐴𝑄𝐴 ) → 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 ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
17 14 oveq2i ( 𝐷 𝑉 ) = ( 𝐷 ( ( 𝑍 𝑆 ) 𝑊 ) )
18 17 a1i ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝐷 𝑉 ) = ( 𝐷 ( ( 𝑍 𝑆 ) 𝑊 ) ) )
19 16 18 oveq12d ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( ( 𝑃 𝑄 ) ( 𝐷 𝑉 ) ) = ( ( 𝑄 𝑃 ) ( 𝐷 ( ( 𝑍 𝑆 ) 𝑊 ) ) ) )
20 12 19 eqtr4id ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐷 𝑉 ) ) )