Metamath Proof Explorer


Theorem cdleme6

Description: Part of proof of Lemma E in Crawley p. 113. This expresses (r \/ f_s(r)) /\ w = u at the top of p. 114. (Contributed by NM, 7-Jun-2012)

Ref Expression
Hypotheses cdleme4.l = ( le ‘ 𝐾 )
cdleme4.j = ( join ‘ 𝐾 )
cdleme4.m = ( meet ‘ 𝐾 )
cdleme4.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme4.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme4.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme4.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme4.g 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑅 𝑆 ) 𝑊 ) ) )
Assertion cdleme6 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( 𝑅 𝐺 ) 𝑊 ) = 𝑈 )

Proof

Step Hyp Ref Expression
1 cdleme4.l = ( le ‘ 𝐾 )
2 cdleme4.j = ( join ‘ 𝐾 )
3 cdleme4.m = ( meet ‘ 𝐾 )
4 cdleme4.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme4.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme4.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme4.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
8 cdleme4.g 𝐺 = ( ( 𝑃 𝑄 ) ( 𝐹 ( ( 𝑅 𝑆 ) 𝑊 ) ) )
9 1 2 3 4 5 6 7 8 cdleme5 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑅 𝐺 ) = ( 𝑃 𝑄 ) )
10 9 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( 𝑅 𝐺 ) 𝑊 ) = ( ( 𝑃 𝑄 ) 𝑊 ) )
11 10 6 eqtr4di ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( 𝑅 𝐺 ) 𝑊 ) = 𝑈 )