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Metamath Proof Explorer

Theorem trlcnv

Description: The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013)

Ref Expression
Hypotheses trlcnv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trlcnv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlcnv.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlcnv ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ◡ 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 trlcnv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 trlcnv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 trlcnv.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
4 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
6 4 5 1 lhpexnle ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 )
7 6 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ∃ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 )
8 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9 8 1 2 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
10 9 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
11 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑝 ∈ ( Atoms ‘ 𝐾 ) )
12 8 5 atbase ⊢ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) → 𝑝 ∈ ( Base ‘ 𝐾 ) )
13 11 12 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑝 ∈ ( Base ‘ 𝐾 ) )
14 f1ocnvfv1 ⊢ ( ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ 𝑝 ∈ ( Base ‘ 𝐾 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑝 ) ) = 𝑝 )
15 10 13 14 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑝 ) ) = 𝑝 )
16 15 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑝 ) ) ) = ( ( 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) 𝑝 ) )
17 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐾 ∈ HL )
18 4 5 1 2 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) )
19 18 3adant3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) )
20 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
21 20 5 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ) → ( ( 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) 𝑝 ) = ( 𝑝 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑝 ) ) )
22 17 19 11 21 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) 𝑝 ) = ( 𝑝 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑝 ) ) )
23 16 22 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑝 ) ) ) = ( 𝑝 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑝 ) ) )
24 23 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ( 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑝 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑝 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
25 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
26 1 2 ltrncnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 )
27 26 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ◡ 𝐹 ∈ 𝑇 )
28 4 5 1 2 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( 𝐹 ‘ 𝑝 ) ( le ‘ 𝐾 ) 𝑊 ) )
29 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
30 4 20 29 5 1 2 3 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ◡ 𝐹 ∈ 𝑇 ∧ ( ( 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( 𝐹 ‘ 𝑝 ) ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑅 ‘ ◡ 𝐹 ) = ( ( ( 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑝 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
31 25 27 28 30 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑅 ‘ ◡ 𝐹 ) = ( ( ( 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ ( 𝐹 ‘ 𝑝 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
32 4 20 29 5 1 2 3 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑝 ( join ‘ 𝐾 ) ( 𝐹 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
33 24 31 32 3eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑅 ‘ ◡ 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )
34 33 3expa ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑅 ‘ ◡ 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )
35 7 34 rexlimddv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ◡ 𝐹 ) = ( 𝑅 ‘ 𝐹 ) )