Metamath Proof Explorer


Theorem cdleme19c

Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114, 1st line. D , F represent s_2, f(s). We prove f(s) =/= s_2. (Contributed by NM, 13-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 )
Assertion cdleme19c ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐹𝐷 )

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 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ HL )
12 11 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ Lat )
13 simp31 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑅𝐴 )
14 simp23l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑆𝐴 )
15 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
16 15 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴 ) → ( 𝑅 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
17 11 13 14 16 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑅 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
18 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑊𝐻 )
19 15 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
20 18 19 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
21 15 1 3 latmle2 ( ( 𝐾 ∈ Lat ∧ ( 𝑅 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑅 𝑆 ) 𝑊 ) 𝑊 )
22 12 17 20 21 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( ( 𝑅 𝑆 ) 𝑊 ) 𝑊 )
23 9 22 eqbrtrid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐷 𝑊 )
24 simp32 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑃𝑄 )
25 simp33 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
26 24 25 jca ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) )
27 1 2 3 4 5 6 7 cdleme3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐹 𝑊 )
28 26 27 syld3an3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐹 𝑊 )
29 nbrne2 ( ( 𝐷 𝑊 ∧ ¬ 𝐹 𝑊 ) → 𝐷𝐹 )
30 23 28 29 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐷𝐹 )
31 30 necomd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐹𝐷 )