Metamath Proof Explorer


Theorem diafval

Description: The partial isomorphism A for a lattice K . (Contributed by NM, 15-Oct-2013)

Ref Expression
Hypotheses diaval.b 𝐵 = ( Base ‘ 𝐾 )
diaval.l = ( le ‘ 𝐾 )
diaval.h 𝐻 = ( LHyp ‘ 𝐾 )
diaval.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
diaval.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
diaval.i 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion diafval ( ( 𝐾𝑉𝑊𝐻 ) → 𝐼 = ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑊 } ↦ { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑥 } ) )

Proof

Step Hyp Ref Expression
1 diaval.b 𝐵 = ( Base ‘ 𝐾 )
2 diaval.l = ( le ‘ 𝐾 )
3 diaval.h 𝐻 = ( LHyp ‘ 𝐾 )
4 diaval.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 diaval.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 diaval.i 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7 1 2 3 diaffval ( 𝐾𝑉 → ( DIsoA ‘ 𝐾 ) = ( 𝑤𝐻 ↦ ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 } ) ) )
8 7 fveq1d ( 𝐾𝑉 → ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤𝐻 ↦ ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 } ) ) ‘ 𝑊 ) )
9 6 8 syl5eq ( 𝐾𝑉𝐼 = ( ( 𝑤𝐻 ↦ ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 } ) ) ‘ 𝑊 ) )
10 breq2 ( 𝑤 = 𝑊 → ( 𝑦 𝑤𝑦 𝑊 ) )
11 10 rabbidv ( 𝑤 = 𝑊 → { 𝑦𝐵𝑦 𝑤 } = { 𝑦𝐵𝑦 𝑊 } )
12 fveq2 ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
13 12 4 eqtr4di ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝑇 )
14 fveq2 ( 𝑤 = 𝑊 → ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) )
15 14 5 eqtr4di ( 𝑤 = 𝑊 → ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) = 𝑅 )
16 15 fveq1d ( 𝑤 = 𝑊 → ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) = ( 𝑅𝑓 ) )
17 16 breq1d ( 𝑤 = 𝑊 → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 ↔ ( 𝑅𝑓 ) 𝑥 ) )
18 13 17 rabeqbidv ( 𝑤 = 𝑊 → { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 } = { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑥 } )
19 11 18 mpteq12dv ( 𝑤 = 𝑊 → ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 } ) = ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑊 } ↦ { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑥 } ) )
20 eqid ( 𝑤𝐻 ↦ ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 } ) ) = ( 𝑤𝐻 ↦ ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 } ) )
21 1 fvexi 𝐵 ∈ V
22 21 mptrabex ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑊 } ↦ { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑥 } ) ∈ V
23 19 20 22 fvmpt ( 𝑊𝐻 → ( ( 𝑤𝐻 ↦ ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) 𝑥 } ) ) ‘ 𝑊 ) = ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑊 } ↦ { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑥 } ) )
24 9 23 sylan9eq ( ( 𝐾𝑉𝑊𝐻 ) → 𝐼 = ( 𝑥 ∈ { 𝑦𝐵𝑦 𝑊 } ↦ { 𝑓𝑇 ∣ ( 𝑅𝑓 ) 𝑥 } ) )