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Metamath Proof Explorer

Theorem ltrncnv

Description: The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013)

Ref Expression
Hypotheses ltrncnv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrncnv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrncnv ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 )

Proof

Step Hyp Ref Expression
1 ltrncnv.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 ltrncnv.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 eqid ⊢ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
4 1 3 2 ltrnldil ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
5 1 3 ldilcnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) → ◡ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
6 4 5 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
7 simp1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) )
8 simp1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp1r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐹 ∈ 𝑇 )
10 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑝 ∈ ( Atoms ‘ 𝐾 ) )
11 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 )
12 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
13 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
14 12 13 1 2 ltrncnvel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑝 ) ( le ‘ 𝐾 ) 𝑊 ) )
15 8 9 10 11 14 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑝 ) ( le ‘ 𝐾 ) 𝑊 ) )
16 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑞 ∈ ( Atoms ‘ 𝐾 ) )
17 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 )
18 12 13 1 2 ltrncnvel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑞 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑞 ) ( le ‘ 𝐾 ) 𝑊 ) )
19 8 9 16 17 18 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑞 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑞 ) ( le ‘ 𝐾 ) 𝑊 ) )
20 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
21 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
22 12 20 21 13 1 2 ltrnu ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( ◡ 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑝 ) ( le ‘ 𝐾 ) 𝑊 ) ∧ ( ( ◡ 𝐹 ‘ 𝑞 ) ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑞 ) ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ( ◡ 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑝 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( ( ◡ 𝐹 ‘ 𝑞 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑞 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
23 7 15 19 22 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ( ◡ 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑝 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( ( ◡ 𝐹 ‘ 𝑞 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑞 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
24 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
25 24 1 2 ltrn1o ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
26 25 3ad2ant1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) )
27 24 13 atbase ⊢ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) → 𝑝 ∈ ( Base ‘ 𝐾 ) )
28 10 27 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑝 ∈ ( Base ‘ 𝐾 ) )
29 f1ocnvfv2 ⊢ ( ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ 𝑝 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑝 ) ) = 𝑝 )
30 26 28 29 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑝 ) ) = 𝑝 )
31 30 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑝 ) ) ) = ( ( ◡ 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) 𝑝 ) )
32 simp1ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐾 ∈ HL )
33 12 13 1 2 ltrncnvat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ) → ( ◡ 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) )
34 8 9 10 33 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ◡ 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) )
35 20 13 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ ( ◡ 𝐹 ‘ 𝑝 ) ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ) → ( ( ◡ 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) 𝑝 ) = ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) )
36 32 34 10 35 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) 𝑝 ) = ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) )
37 31 36 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑝 ) ) ) = ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) )
38 37 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ( ◡ 𝐹 ‘ 𝑝 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑝 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
39 24 13 atbase ⊢ ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) → 𝑞 ∈ ( Base ‘ 𝐾 ) )
40 16 39 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑞 ∈ ( Base ‘ 𝐾 ) )
41 f1ocnvfv2 ⊢ ( ( 𝐹 : ( Base ‘ 𝐾 ) –1-1-onto→ ( Base ‘ 𝐾 ) ∧ 𝑞 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑞 ) ) = 𝑞 )
42 26 40 41 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑞 ) ) = 𝑞 )
43 42 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑞 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑞 ) ) ) = ( ( ◡ 𝐹 ‘ 𝑞 ) ( join ‘ 𝐾 ) 𝑞 ) )
44 12 13 1 2 ltrncnvat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) → ( ◡ 𝐹 ‘ 𝑞 ) ∈ ( Atoms ‘ 𝐾 ) )
45 8 9 16 44 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ◡ 𝐹 ‘ 𝑞 ) ∈ ( Atoms ‘ 𝐾 ) )
46 20 13 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ ( ◡ 𝐹 ‘ 𝑞 ) ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) → ( ( ◡ 𝐹 ‘ 𝑞 ) ( join ‘ 𝐾 ) 𝑞 ) = ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) )
47 32 45 16 46 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑞 ) ( join ‘ 𝐾 ) 𝑞 ) = ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) )
48 43 47 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ◡ 𝐹 ‘ 𝑞 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑞 ) ) ) = ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) )
49 48 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( ( ◡ 𝐹 ‘ 𝑞 ) ( join ‘ 𝐾 ) ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑞 ) ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
50 23 38 49 3eqtr3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
51 50 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) → ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
52 51 ralrimivv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
53 12 20 21 13 1 3 2 isltrn ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ◡ 𝐹 ∈ 𝑇 ↔ ( ◡ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) )
54 53 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ◡ 𝐹 ∈ 𝑇 ↔ ( ◡ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ◡ 𝐹 ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) )
55 6 52 54 mpbir2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 )