Metamath Proof Explorer


Theorem cdleme20bN

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, second line. D , F , Y , G represent s_2, f(s), t_2, f(t). We show v \/ s_2 = v \/ t_2. (Contributed by NM, 15-Nov-2012) (New usage is discouraged.)

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 )
cdleme20.v 𝑉 = ( ( 𝑆 𝑇 ) 𝑊 )
Assertion cdleme20bN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑉 𝐷 ) = ( 𝑉 𝑌 ) )

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 cdleme20.v 𝑉 = ( ( 𝑆 𝑇 ) 𝑊 )
12 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ HL )
13 12 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ Lat )
14 simp22l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑆𝐴 )
15 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
16 15 4 atbase ( 𝑆𝐴𝑆 ∈ ( Base ‘ 𝐾 ) )
17 14 16 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑆 ∈ ( Base ‘ 𝐾 ) )
18 simp21 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑅𝐴 )
19 15 4 atbase ( 𝑅𝐴𝑅 ∈ ( Base ‘ 𝐾 ) )
20 18 19 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
21 simp23l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑇𝐴 )
22 15 4 atbase ( 𝑇𝐴𝑇 ∈ ( Base ‘ 𝐾 ) )
23 21 22 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑇 ∈ ( Base ‘ 𝐾 ) )
24 15 2 latj31 ( ( 𝐾 ∈ Lat ∧ ( 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ 𝑇 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑆 𝑅 ) 𝑇 ) = ( ( 𝑇 𝑅 ) 𝑆 ) )
25 13 17 20 23 24 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( 𝑆 𝑅 ) 𝑇 ) = ( ( 𝑇 𝑅 ) 𝑆 ) )
26 25 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( ( 𝑆 𝑅 ) 𝑇 ) 𝑊 ) = ( ( ( 𝑇 𝑅 ) 𝑆 ) 𝑊 ) )
27 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑊𝐻 )
28 simp22r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ¬ 𝑆 𝑊 )
29 simp31 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
30 simp33 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑅 ( 𝑃 𝑄 ) )
31 1 2 3 4 5 6 7 8 9 10 11 cdleme20aN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑉 𝐷 ) = ( ( ( 𝑆 𝑅 ) 𝑇 ) 𝑊 ) )
32 12 27 18 14 28 21 29 30 31 syl233anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑉 𝐷 ) = ( ( ( 𝑆 𝑅 ) 𝑇 ) 𝑊 ) )
33 2 4 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴 ) → ( 𝑆 𝑇 ) = ( 𝑇 𝑆 ) )
34 12 14 21 33 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑆 𝑇 ) = ( 𝑇 𝑆 ) )
35 34 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( 𝑆 𝑇 ) 𝑊 ) = ( ( 𝑇 𝑆 ) 𝑊 ) )
36 11 35 syl5eq ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑉 = ( ( 𝑇 𝑆 ) 𝑊 ) )
37 36 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑉 𝑌 ) = ( ( ( 𝑇 𝑆 ) 𝑊 ) 𝑌 ) )
38 simp23r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ¬ 𝑇 𝑊 )
39 simp32 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ¬ 𝑇 ( 𝑃 𝑄 ) )
40 eqid ( ( 𝑇 𝑆 ) 𝑊 ) = ( ( 𝑇 𝑆 ) 𝑊 )
41 1 2 3 4 5 6 8 7 10 9 40 cdleme20aN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( ( 𝑇 𝑆 ) 𝑊 ) 𝑌 ) = ( ( ( 𝑇 𝑅 ) 𝑆 ) 𝑊 ) )
42 12 27 18 21 38 14 39 30 41 syl233anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( ( 𝑇 𝑆 ) 𝑊 ) 𝑌 ) = ( ( ( 𝑇 𝑅 ) 𝑆 ) 𝑊 ) )
43 37 42 eqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑉 𝑌 ) = ( ( ( 𝑇 𝑅 ) 𝑆 ) 𝑊 ) )
44 26 32 43 3eqtr4d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑉 𝐷 ) = ( 𝑉 𝑌 ) )