# Metamath Proof Explorer

## Theorem ltrnel

Description: The lattice translation of an atom not under the fiducial co-atom is also an atom not under the fiducial co-atom. Remark below Lemma B in Crawley p. 112. (Contributed by NM, 22-May-2012)

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

### Proof

Step Hyp Ref Expression
1 ltrnel.l = ( le ‘ 𝐾 )
2 ltrnel.a 𝐴 = ( Atoms ‘ 𝐾 )
3 ltrnel.h 𝐻 = ( LHyp ‘ 𝐾 )
4 ltrnel.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑃𝐴 )
6 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7 6 2 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
8 7 adantr ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
9 6 2 3 4 ltrnatb ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃𝐴 ↔ ( 𝐹𝑃 ) ∈ 𝐴 ) )
10 8 9 syl3an3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃𝐴 ↔ ( 𝐹𝑃 ) ∈ 𝐴 ) )
11 5 10 mpbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐹𝑃 ) ∈ 𝐴 )
12 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ¬ 𝑃 𝑊 )
13 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 simp2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐹𝑇 )
15 5 7 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
16 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑊𝐻 )
17 6 3 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
18 16 17 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
19 6 1 3 4 ltrnle ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑃 𝑊 ↔ ( 𝐹𝑃 ) ( 𝐹𝑊 ) ) )
20 13 14 15 18 19 syl112anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 𝑊 ↔ ( 𝐹𝑃 ) ( 𝐹𝑊 ) ) )
21 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐾 ∈ HL )
22 21 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐾 ∈ Lat )
23 6 1 latref ( ( 𝐾 ∈ Lat ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → 𝑊 𝑊 )
24 22 18 23 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑊 𝑊 )
25 6 1 3 4 ltrnval1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 𝑊 ) ) → ( 𝐹𝑊 ) = 𝑊 )
26 13 14 18 24 25 syl112anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝐹𝑊 ) = 𝑊 )
27 26 breq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐹𝑃 ) ( 𝐹𝑊 ) ↔ ( 𝐹𝑃 ) 𝑊 ) )
28 20 27 bitrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 𝑊 ↔ ( 𝐹𝑃 ) 𝑊 ) )
29 12 28 mtbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ¬ ( 𝐹𝑃 ) 𝑊 )
30 11 29 jca ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐹𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑃 ) 𝑊 ) )