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Metamath Proof Explorer

Theorem trlle

Description: The trace of a lattice translation is less than the fiducial co-atom W . (Contributed by NM, 25-May-2012)

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

Proof

Step Hyp Ref Expression
1 trlle.l ⊢ ≤ = ( le ‘ 𝐾 )
2 trlle.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 trlle.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 trlle.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
6 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
7 1 5 6 2 lhpocnel ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) )
8 7 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) )
9 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
10 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
11 1 9 10 6 2 3 4 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
12 8 11 mpd3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
13 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
14 13 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ Lat )
15 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
16 15 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐾 ∈ OP )
17 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
18 17 2 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
19 18 ad2antlr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
20 17 5 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
21 16 19 20 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
22 17 2 3 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) )
23 21 22 mpd3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) )
24 17 9 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∈ ( Base ‘ 𝐾 ) )
25 14 21 23 24 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∈ ( Base ‘ 𝐾 ) )
26 17 1 10 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 )
27 14 25 19 26 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 )
28 12 27 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 )