Metamath Proof Explorer


Theorem ltrnset

Description: The set of lattice translations for a fiducial co-atom W . (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 ltrnset ( ( 𝐾𝐵𝑊𝐻 ) → 𝑇 = { 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 ltrnfset ( 𝐾𝐵 → ( LTrn ‘ 𝐾 ) = ( 𝑤𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) } ) )
9 8 fveq1d ( 𝐾𝐵 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) } ) ‘ 𝑊 ) )
10 7 9 syl5eq ( 𝐾𝐵𝑇 = ( ( 𝑤𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) } ) ‘ 𝑊 ) )
11 fveq2 ( 𝑤 = 𝑊 → ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
12 11 6 eqtr4di ( 𝑤 = 𝑊 → ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) = 𝐷 )
13 breq2 ( 𝑤 = 𝑊 → ( 𝑝 𝑤𝑝 𝑊 ) )
14 13 notbid ( 𝑤 = 𝑊 → ( ¬ 𝑝 𝑤 ↔ ¬ 𝑝 𝑊 ) )
15 breq2 ( 𝑤 = 𝑊 → ( 𝑞 𝑤𝑞 𝑊 ) )
16 15 notbid ( 𝑤 = 𝑊 → ( ¬ 𝑞 𝑤 ↔ ¬ 𝑞 𝑊 ) )
17 14 16 anbi12d ( 𝑤 = 𝑊 → ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) ↔ ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) ) )
18 oveq2 ( 𝑤 = 𝑊 → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) )
19 oveq2 ( 𝑤 = 𝑊 → ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) )
20 18 19 eqeq12d ( 𝑤 = 𝑊 → ( ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ↔ ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) )
21 17 20 imbi12d ( 𝑤 = 𝑊 → ( ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) ↔ ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) ) )
22 21 2ralbidv ( 𝑤 = 𝑊 → ( ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) ↔ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) ) )
23 12 22 rabeqbidv ( 𝑤 = 𝑊 → { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) } = { 𝑓𝐷 ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) } )
24 eqid ( 𝑤𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) } ) = ( 𝑤𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) } )
25 6 fvexi 𝐷 ∈ V
26 25 rabex { 𝑓𝐷 ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) } ∈ V
27 23 24 26 fvmpt ( 𝑊𝐻 → ( ( 𝑤𝐻 ↦ { 𝑓 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑤 ) ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑤 ∧ ¬ 𝑞 𝑤 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑤 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑤 ) ) } ) ‘ 𝑊 ) = { 𝑓𝐷 ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) } )
28 10 27 sylan9eq ( ( 𝐾𝐵𝑊𝐻 ) → 𝑇 = { 𝑓𝐷 ∣ ∀ 𝑝𝐴𝑞𝐴 ( ( ¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊 ) → ( ( 𝑝 ( 𝑓𝑝 ) ) 𝑊 ) = ( ( 𝑞 ( 𝑓𝑞 ) ) 𝑊 ) ) } )