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Metamath Proof Explorer

Theorem trlval3

Description: The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013)

Ref Expression
Hypotheses trlval3.l ⊢ ≤ = ( le ‘ 𝐾 )
trlval3.j ⊢ ∨ = ( join ‘ 𝐾 )
trlval3.m ⊢ ∧ = ( meet ‘ 𝐾 )
trlval3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
trlval3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trlval3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlval3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlval3 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )

Proof

Step Hyp Ref Expression
1 trlval3.l ⊢ ≤ = ( le ‘ 𝐾 )
2 trlval3.j ⊢ ∨ = ( join ‘ 𝐾 )
3 trlval3.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 trlval3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 trlval3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 trlval3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 trlval3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simpl31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
10 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → 𝐹 ∈ 𝑇 )
11 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 )
12 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
13 1 12 4 5 6 7 trl0 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) )
14 8 9 10 11 13 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) )
15 simpl33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
16 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → 𝐾 ∈ HL )
17 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
18 16 17 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → 𝐾 ∈ AtLat )
19 11 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑃 ) )
20 simp31l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ∈ 𝐴 )
21 20 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → 𝑃 ∈ 𝐴 )
22 2 4 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑃 ) = 𝑃 )
23 16 21 22 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑃 ∨ 𝑃 ) = 𝑃 )
24 19 23 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = 𝑃 )
25 24 21 eqeltrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ 𝐴 )
26 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
27 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝐹 ∈ 𝑇 )
28 simp31 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
29 simp32 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
30 1 4 5 6 ltrn2ateq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) = 𝑃 ↔ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) )
31 26 27 28 29 30 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) = 𝑃 ↔ ( 𝐹 ‘ 𝑄 ) = 𝑄 ) )
32 31 biimpa ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐹 ‘ 𝑄 ) = 𝑄 )
33 32 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ 𝑄 ) )
34 simp32l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑄 ∈ 𝐴 )
35 34 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → 𝑄 ∈ 𝐴 )
36 2 4 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑄 ) = 𝑄 )
37 16 35 36 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑄 ∨ 𝑄 ) = 𝑄 )
38 33 37 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) = 𝑄 )
39 38 35 eqeltrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ 𝐴 )
40 3 12 4 atnem0 ⊢ ( ( 𝐾 ∈ AtLat ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ 𝐴 ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ 𝐴 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ↔ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) = ( 0. ‘ 𝐾 ) ) )
41 18 25 39 40 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ↔ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) = ( 0. ‘ 𝐾 ) ) )
42 15 41 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) = ( 0. ‘ 𝐾 ) )
43 14 42 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
44 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
45 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝐹 ∈ 𝑇 )
46 simpl31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
47 1 2 3 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) )
48 44 45 46 47 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) )
49 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝐾 ∈ HL )
50 49 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝐾 ∈ Lat )
51 20 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝑃 ∈ 𝐴 )
52 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
53 44 45 51 52 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
54 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
55 54 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
56 49 51 53 55 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
57 simpl1r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝑊 ∈ 𝐻 )
58 54 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
59 57 58 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
60 54 1 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
61 50 56 59 60 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
62 48 61 eqbrtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
63 simpl32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
64 1 2 3 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) )
65 44 45 63 64 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) )
66 34 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝑄 ∈ 𝐴 )
67 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
68 44 45 66 67 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
69 54 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
70 49 66 68 69 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
71 54 1 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
72 50 70 59 71 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑊 ) ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
73 65 72 eqbrtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
74 54 5 6 7 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
75 44 45 74 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
76 54 1 3 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) )
77 50 75 56 70 76 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ( ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) )
78 62 73 77 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
79 49 17 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → 𝐾 ∈ AtLat )
80 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 )
81 1 4 5 6 7 trlat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )
82 44 46 45 80 81 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )
83 54 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) )
84 50 56 70 83 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) )
85 54 1 12 4 atlen0 ⊢ ( ( ( 𝐾 ∈ AtLat ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ) ∧ ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ≠ ( 0. ‘ 𝐾 ) )
86 79 84 82 78 85 syl31anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ≠ ( 0. ‘ 𝐾 ) )
87 86 neneqd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ¬ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) = ( 0. ‘ 𝐾 ) )
88 simpl33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
89 2 3 12 4 2atmat0 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∈ 𝐴 ∨ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) = ( 0. ‘ 𝐾 ) ) )
90 49 51 53 66 68 88 89 syl33anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∈ 𝐴 ∨ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) = ( 0. ‘ 𝐾 ) ) )
91 90 ord ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ¬ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∈ 𝐴 → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) = ( 0. ‘ 𝐾 ) ) )
92 87 91 mt3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∈ 𝐴 )
93 1 4 atcmp ⊢ ( ( 𝐾 ∈ AtLat ∧ ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∈ 𝐴 ) → ( ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ↔ ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) )
94 79 82 92 93 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ↔ ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) )
95 78 94 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
96 43 95 pm2.61dane ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )