Metamath Proof Explorer


Theorem isltrn

Description: The predicate "is a lattice translation". Similar to definition of translation in Crawley p. 111. (Contributed by NM, 11-May-2012)

Ref Expression
Hypotheses ltrnset.l = ( le ‘ 𝐾 )
ltrnset.j = ( join ‘ 𝐾 )
ltrnset.m = ( meet ‘ 𝐾 )
ltrnset.a 𝐴 = ( Atoms ‘ 𝐾 )
ltrnset.h 𝐻 = ( LHyp ‘ 𝐾 )
ltrnset.d 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
ltrnset.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion isltrn ( ( 𝐾𝐵𝑊𝐻 ) → ( 𝐹𝑇 ↔ ( 𝐹𝐷 ∧ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝐹𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝐹𝑞 ) ) 𝑊 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 ltrnset.l = ( le ‘ 𝐾 )
2 ltrnset.j = ( join ‘ 𝐾 )
3 ltrnset.m = ( meet ‘ 𝐾 )
4 ltrnset.a 𝐴 = ( Atoms ‘ 𝐾 )
5 ltrnset.h 𝐻 = ( LHyp ‘ 𝐾 )
6 ltrnset.d 𝐷 = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
7 ltrnset.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
8 1 2 3 4 5 6 7 ltrnset ( ( 𝐾𝐵𝑊𝐻 ) → 𝑇 = { 𝑓𝐷 ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) } )
9 8 eleq2d ( ( 𝐾𝐵𝑊𝐻 ) → ( 𝐹𝑇𝐹 ∈ { 𝑓𝐷 ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) } ) )
10 fveq1 ( 𝑓 = 𝐹 → ( 𝑓𝑝 ) = ( 𝐹𝑝 ) )
11 10 oveq2d ( 𝑓 = 𝐹 → ( 𝑝 ( 𝑓𝑝 ) ) = ( 𝑝 ( 𝐹𝑝 ) ) )
12 11 oveq1d ( 𝑓 = 𝐹 → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑝 ( 𝐹𝑝 ) ) 𝑊 ) )
13 fveq1 ( 𝑓 = 𝐹 → ( 𝑓𝑞 ) = ( 𝐹𝑞 ) )
14 13 oveq2d ( 𝑓 = 𝐹 → ( 𝑞 ( 𝑓𝑞 ) ) = ( 𝑞 ( 𝐹𝑞 ) ) )
15 14 oveq1d ( 𝑓 = 𝐹 → ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) = ( ( 𝑞 ( 𝐹𝑞 ) ) 𝑊 ) )
16 12 15 eqeq12d ( 𝑓 = 𝐹 → ( ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ↔ ( ( 𝑝 ( 𝐹𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝐹𝑞 ) ) 𝑊 ) ) )
17 16 imbi2d ( 𝑓 = 𝐹 → ( ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) ↔ ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝐹𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝐹𝑞 ) ) 𝑊 ) ) ) )
18 17 2ralbidv ( 𝑓 = 𝐹 → ( ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) ↔ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝐹𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝐹𝑞 ) ) 𝑊 ) ) ) )
19 18 elrab ( 𝐹 ∈ { 𝑓𝐷 ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) } ↔ ( 𝐹𝐷 ∧ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝐹𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝐹𝑞 ) ) 𝑊 ) ) ) )
20 9 19 bitrdi ( ( 𝐾𝐵𝑊𝐻 ) → ( 𝐹𝑇 ↔ ( 𝐹𝐷 ∧ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝐹𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝐹𝑞 ) ) 𝑊 ) ) ) ) )