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Metamath Proof Explorer

Theorem trlcl

Description: Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012)

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

Proof

Step Hyp Ref Expression
1 trlcl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 trlcl.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 trlcl.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 trlcl.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6 eqid ⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
7 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
8 5 6 7 2 lhpocnel ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) )
9 8 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) )
10 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
11 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
12 5 10 11 7 2 3 4 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
13 9 12 mpd3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
14 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
15 14 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ Lat )
16 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
17 16 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ OP )
18 1 2 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
19 18 ad2antlr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑊 ∈ 𝐵 )
20 1 6 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ 𝐵 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐵 )
21 17 19 20 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐵 )
22 1 2 3 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐵 ) → ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ 𝐵 )
23 21 22 mpd3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ 𝐵 )
24 1 10 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ 𝐵 ∧ ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ 𝐵 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∈ 𝐵 )
25 15 21 23 24 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∈ 𝐵 )
26 1 11 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
27 15 25 19 26 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ 𝐵 )
28 13 27 eqeltrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐵 )