Metamath Proof Explorer


Theorem cdleme19f

Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114, line 3. D , F , N , Y , G , O represent s_2, f(s), f_s(r), t_2, f(t), f_t(r). We prove that if r <_ s \/ t, then f_t(r) = f_t(r). (Contributed by NM, 14-Nov-2012)

Ref Expression
Hypotheses cdleme19.l = ( le ‘ 𝐾 )
cdleme19.j = ( join ‘ 𝐾 )
cdleme19.m = ( meet ‘ 𝐾 )
cdleme19.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme19.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme19.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme19.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme19.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
cdleme19.d 𝐷 = ( ( 𝑅 𝑆 ) 𝑊 )
cdleme19.y 𝑌 = ( ( 𝑅 𝑇 ) 𝑊 )
cdleme19.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 𝐷 ) )
cdleme19.o 𝑂 = ( ( 𝑃 𝑄 ) ( 𝐺 𝑌 ) )
Assertion cdleme19f ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝑁 = 𝑂 )

Proof

Step Hyp Ref Expression
1 cdleme19.l = ( le ‘ 𝐾 )
2 cdleme19.j = ( join ‘ 𝐾 )
3 cdleme19.m = ( meet ‘ 𝐾 )
4 cdleme19.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme19.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme19.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme19.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
8 cdleme19.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
9 cdleme19.d 𝐷 = ( ( 𝑅 𝑆 ) 𝑊 )
10 cdleme19.y 𝑌 = ( ( 𝑅 𝑇 ) 𝑊 )
11 cdleme19.n 𝑁 = ( ( 𝑃 𝑄 ) ( 𝐹 𝐷 ) )
12 cdleme19.o 𝑂 = ( ( 𝑃 𝑄 ) ( 𝐺 𝑌 ) )
13 1 2 3 4 5 6 7 8 9 10 cdleme19e ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝐹 𝐷 ) = ( 𝐺 𝑌 ) )
14 13 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( ( 𝑃 𝑄 ) ( 𝐹 𝐷 ) ) = ( ( 𝑃 𝑄 ) ( 𝐺 𝑌 ) ) )
15 14 11 12 3eqtr4g ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝑁 = 𝑂 )