Metamath Proof Explorer


Theorem trljat2

Description: The value of a translation of an atom P not under the fiducial co-atom W , joined with trace. Equation above Lemma C in Crawley p. 112. (Contributed by NM, 25-May-2012)

Ref Expression
Hypotheses trljat.l = ( le ‘ 𝐾 )
trljat.j = ( join ‘ 𝐾 )
trljat.a 𝐴 = ( Atoms ‘ 𝐾 )
trljat.h 𝐻 = ( LHyp ‘ 𝐾 )
trljat.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trljat.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trljat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐹𝑃 ) ( 𝑅𝐹 ) ) = ( 𝑃 ( 𝐹𝑃 ) ) )

Proof

Step Hyp Ref Expression
1 trljat.l = ( le ‘ 𝐾 )
2 trljat.j = ( join ‘ 𝐾 )
3 trljat.a 𝐴 = ( Atoms ‘ 𝐾 )
4 trljat.h 𝐻 = ( LHyp ‘ 𝐾 )
5 trljat.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 trljat.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐾 ∈ HL )
8 1 3 4 5 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝑃𝐴 ) → ( 𝐹𝑃 ) ∈ 𝐴 )
9 8 3adant3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐹𝑃 ) ∈ 𝐴 )
10 7 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐾 ∈ Lat )
11 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑃𝐴 )
12 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
13 12 3 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
14 11 13 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
15 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
16 simp2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐹𝑇 )
17 12 4 5 ltrncl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹𝑃 ) ∈ ( Base ‘ 𝐾 ) )
18 15 16 14 17 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐹𝑃 ) ∈ ( Base ‘ 𝐾 ) )
19 12 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ( 𝐹𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
20 10 14 18 19 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 ( 𝐹𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
21 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑊𝐻 )
22 12 4 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
23 21 22 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
24 12 1 2 latlej2 ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹𝑃 ) ( 𝑃 ( 𝐹𝑃 ) ) )
25 10 14 18 24 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐹𝑃 ) ( 𝑃 ( 𝐹𝑃 ) ) )
26 eqid ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
27 12 1 2 26 3 atmod2i1 ( ( 𝐾 ∈ HL ∧ ( ( 𝐹𝑃 ) ∈ 𝐴 ∧ ( 𝑃 ( 𝐹𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝐹𝑃 ) ( 𝑃 ( 𝐹𝑃 ) ) ) → ( ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( 𝐹𝑃 ) ) = ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ( 𝐹𝑃 ) ) ) )
28 7 9 20 23 25 27 syl131anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( 𝐹𝑃 ) ) = ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ( 𝐹𝑃 ) ) ) )
29 1 3 4 5 ltrnel ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐹𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑃 ) 𝑊 ) )
30 eqid ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
31 1 2 30 3 4 lhpjat1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝐹𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑃 ) 𝑊 ) ) → ( 𝑊 ( 𝐹𝑃 ) ) = ( 1. ‘ 𝐾 ) )
32 7 21 29 31 syl21anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑊 ( 𝐹𝑃 ) ) = ( 1. ‘ 𝐾 ) )
33 32 oveq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) ( 𝑊 ( 𝐹𝑃 ) ) ) = ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) )
34 hlol ( 𝐾 ∈ HL → 𝐾 ∈ OL )
35 7 34 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐾 ∈ OL )
36 12 26 30 olm11 ( ( 𝐾 ∈ OL ∧ ( 𝑃 ( 𝐹𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 ( 𝐹𝑃 ) ) )
37 35 20 36 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 ( 𝐹𝑃 ) ) )
38 28 33 37 3eqtrrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 ( 𝐹𝑃 ) ) = ( ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( 𝐹𝑃 ) ) )
39 1 2 26 3 4 5 6 trlval2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑅𝐹 ) = ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
40 39 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝑅𝐹 ) ( 𝐹𝑃 ) ) = ( ( ( 𝑃 ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( 𝐹𝑃 ) ) )
41 12 4 5 6 trlcl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ) → ( 𝑅𝐹 ) ∈ ( Base ‘ 𝐾 ) )
42 15 16 41 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑅𝐹 ) ∈ ( Base ‘ 𝐾 ) )
43 12 2 latjcom ( ( 𝐾 ∈ Lat ∧ ( 𝑅𝐹 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑅𝐹 ) ( 𝐹𝑃 ) ) = ( ( 𝐹𝑃 ) ( 𝑅𝐹 ) ) )
44 10 42 18 43 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝑅𝐹 ) ( 𝐹𝑃 ) ) = ( ( 𝐹𝑃 ) ( 𝑅𝐹 ) ) )
45 38 40 44 3eqtr2rd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐹𝑃 ) ( 𝑅𝐹 ) ) = ( 𝑃 ( 𝐹𝑃 ) ) )