Metamath Proof Explorer


Theorem trl0

Description: If an atom not under the fiducial co-atom W equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012)

Ref Expression
Hypotheses trl0.l = ( le ‘ 𝐾 )
trl0.z 0 = ( 0. ‘ 𝐾 )
trl0.a 𝐴 = ( Atoms ‘ 𝐾 )
trl0.h 𝐻 = ( LHyp ‘ 𝐾 )
trl0.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trl0.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trl0 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑅𝐹 ) = 0 )

Proof

Step Hyp Ref Expression
1 trl0.l = ( le ‘ 𝐾 )
2 trl0.z 0 = ( 0. ‘ 𝐾 )
3 trl0.a 𝐴 = ( Atoms ‘ 𝐾 )
4 trl0.h 𝐻 = ( LHyp ‘ 𝐾 )
5 trl0.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 trl0.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
8 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → 𝐹𝑇 )
9 simp2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
10 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
11 eqid ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
12 1 10 11 3 4 5 6 trlval2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑅𝐹 ) = ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
13 7 8 9 12 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑅𝐹 ) = ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
14 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝐹𝑃 ) = 𝑃 )
15 14 oveq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹𝑃 ) ) = ( 𝑃 ( join ‘ 𝐾 ) 𝑃 ) )
16 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → 𝐾 ∈ HL )
17 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → 𝑃𝐴 )
18 10 3 hlatjidm ( ( 𝐾 ∈ HL ∧ 𝑃𝐴 ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑃 ) = 𝑃 )
19 16 17 18 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑃 ( join ‘ 𝐾 ) 𝑃 ) = 𝑃 )
20 15 19 eqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹𝑃 ) ) = 𝑃 )
21 20 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( ( 𝑃 ( join ‘ 𝐾 ) ( 𝐹𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( 𝑃 ( meet ‘ 𝐾 ) 𝑊 ) )
22 1 11 2 3 4 lhpmat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 ( meet ‘ 𝐾 ) 𝑊 ) = 0 )
23 7 9 22 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑃 ( meet ‘ 𝐾 ) 𝑊 ) = 0 )
24 13 21 23 3eqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑅𝐹 ) = 0 )