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Metamath Proof Explorer

Theorem ltrncnvel

Description: The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013)

Ref Expression
Hypotheses ltrnel.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrnel.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrnel.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrnel.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrncnvel ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( ◡ 𝐹 ‘ 𝑃 ) ≤ 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 1 2 3 4 ltrncnvat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
6 5 3adant3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
7 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ 𝑃 ≤ 𝑊 )
8 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
10 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
11 10 2 atbase ⊢ ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 → ( ◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
12 6 11 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
13 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐻 )
14 10 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
15 13 14 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
16 10 1 3 4 ltrnle ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ) )
17 8 9 12 15 16 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ) )
18 10 3 4 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
19 18 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
20 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 )
21 10 2 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
22 20 21 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
23 f1ocnvfv2 ⊢ ( ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) = 𝑃 )
24 19 22 23 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) = 𝑃 )
25 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
26 25 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
27 10 1 latref ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → 𝑊 ≤ 𝑊 )
28 26 15 27 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ≤ 𝑊 )
29 10 1 3 4 ltrnval1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑊 ) = 𝑊 )
30 8 9 15 28 29 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑊 ) = 𝑊 )
31 24 30 breq12d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑃 ) ) ≤ ( 𝐹 ‘ 𝑊 ) ↔ 𝑃 ≤ 𝑊 ) )
32 17 31 bitrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ↔ 𝑃 ≤ 𝑊 ) )
33 7 32 mtbird ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 )
34 6 33 jca ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( ◡ 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )