Metamath Proof Explorer


Theorem cdleme11g

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 14-Jun-2012)

Ref Expression
Hypotheses cdleme11.l = ( le ‘ 𝐾 )
cdleme11.j = ( join ‘ 𝐾 )
cdleme11.m = ( meet ‘ 𝐾 )
cdleme11.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme11.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme11.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme11.c 𝐶 = ( ( 𝑃 𝑆 ) 𝑊 )
cdleme11.d 𝐷 = ( ( 𝑃 𝑇 ) 𝑊 )
cdleme11.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
Assertion cdleme11g ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 𝐹 ) = ( 𝑄 𝐶 ) )

Proof

Step Hyp Ref Expression
1 cdleme11.l = ( le ‘ 𝐾 )
2 cdleme11.j = ( join ‘ 𝐾 )
3 cdleme11.m = ( meet ‘ 𝐾 )
4 cdleme11.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme11.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme11.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme11.c 𝐶 = ( ( 𝑃 𝑆 ) 𝑊 )
8 cdleme11.d 𝐷 = ( ( 𝑃 𝑇 ) 𝑊 )
9 cdleme11.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
10 9 oveq2i ( 𝑄 𝐹 ) = ( 𝑄 ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) )
11 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝐾 ∈ HL )
12 simp22l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑄𝐴 )
13 11 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝐾 ∈ Lat )
14 simp23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑆𝐴 )
15 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
16 15 4 atbase ( 𝑆𝐴𝑆 ∈ ( Base ‘ 𝐾 ) )
17 14 16 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑆 ∈ ( Base ‘ 𝐾 ) )
18 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
19 simp21 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑃𝐴 )
20 1 2 3 4 5 6 15 cdleme0aa ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑃𝐴𝑄𝐴 ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
21 18 19 12 20 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
22 15 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑆 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
23 13 17 21 22 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑆 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
24 15 4 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
25 12 24 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
26 15 4 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
27 19 26 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
28 15 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
29 13 27 17 28 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) )
30 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑊𝐻 )
31 15 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
32 30 31 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
33 15 3 latmcl ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑆 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
34 13 29 32 33 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( ( 𝑃 𝑆 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
35 15 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 𝑆 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) )
36 13 25 34 35 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) )
37 15 1 2 latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 𝑆 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → 𝑄 ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
38 13 25 34 37 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → 𝑄 ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
39 15 1 2 3 4 atmod1i1 ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴 ∧ ( 𝑆 𝑈 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑄 ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) → ( 𝑄 ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) ) = ( ( 𝑄 ( 𝑆 𝑈 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) )
40 11 12 23 36 38 39 syl131anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) ) = ( ( 𝑄 ( 𝑆 𝑈 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) )
41 10 40 syl5eq ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 𝐹 ) = ( ( 𝑄 ( 𝑆 𝑈 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) )
42 simp22 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
43 1 2 3 4 5 6 cdleme0cq ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ) → ( 𝑄 𝑈 ) = ( 𝑃 𝑄 ) )
44 18 19 42 43 syl12anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 𝑈 ) = ( 𝑃 𝑄 ) )
45 44 oveq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑆 ( 𝑄 𝑈 ) ) = ( 𝑆 ( 𝑃 𝑄 ) ) )
46 15 2 latj12 ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑄 ( 𝑆 𝑈 ) ) = ( 𝑆 ( 𝑄 𝑈 ) ) )
47 13 25 17 21 46 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 ( 𝑆 𝑈 ) ) = ( 𝑆 ( 𝑄 𝑈 ) ) )
48 15 2 latj13 ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑄 ( 𝑃 𝑆 ) ) = ( 𝑆 ( 𝑃 𝑄 ) ) )
49 13 25 27 17 48 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 ( 𝑃 𝑆 ) ) = ( 𝑆 ( 𝑃 𝑄 ) ) )
50 45 47 49 3eqtr4d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 ( 𝑆 𝑈 ) ) = ( 𝑄 ( 𝑃 𝑆 ) ) )
51 50 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( ( 𝑄 ( 𝑆 𝑈 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) = ( ( 𝑄 ( 𝑃 𝑆 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) )
52 15 1 3 latmle1 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑆 ) 𝑊 ) ( 𝑃 𝑆 ) )
53 13 29 32 52 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( ( 𝑃 𝑆 ) 𝑊 ) ( 𝑃 𝑆 ) )
54 15 1 2 latjlej2 ( ( 𝐾 ∈ Lat ∧ ( ( ( 𝑃 𝑆 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑃 𝑆 ) 𝑊 ) ( 𝑃 𝑆 ) → ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ( 𝑄 ( 𝑃 𝑆 ) ) ) )
55 13 34 29 25 54 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( ( ( 𝑃 𝑆 ) 𝑊 ) ( 𝑃 𝑆 ) → ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ( 𝑄 ( 𝑃 𝑆 ) ) ) )
56 53 55 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ( 𝑄 ( 𝑃 𝑆 ) ) )
57 15 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ( 𝑃 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) )
58 13 25 29 57 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 ( 𝑃 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) )
59 15 1 3 latleeqm2 ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ( 𝑃 𝑆 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ( 𝑄 ( 𝑃 𝑆 ) ) ↔ ( ( 𝑄 ( 𝑃 𝑆 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) = ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) )
60 13 36 58 59 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ( 𝑄 ( 𝑃 𝑆 ) ) ↔ ( ( 𝑄 ( 𝑃 𝑆 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) = ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) )
61 56 60 mpbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( ( 𝑄 ( 𝑃 𝑆 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) = ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
62 7 oveq2i ( 𝑄 𝐶 ) = ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) )
63 61 62 eqtr4di ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( ( 𝑄 ( 𝑃 𝑆 ) ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) ) = ( 𝑄 𝐶 ) )
64 41 51 63 3eqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑆𝐴 ) ∧ 𝑃𝑄 ) → ( 𝑄 𝐹 ) = ( 𝑄 𝐶 ) )