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 ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 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 ⊢ ( 𝐾 ∈ 𝑉 → ( DIsoA ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) )
8 7 fveq1d ⊢ ( 𝐾 ∈ 𝑉 → ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) ‘ 𝑊 ) )
9 6 8 syl5eq ⊢ ( 𝐾 ∈ 𝑉 → 𝐼 = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) ‘ 𝑊 ) )
10 breq2 ⊢ ( 𝑤 = 𝑊 → ( 𝑦 ≤ 𝑤 ↔ 𝑦 ≤ 𝑊 ) )
11 10 rabbidv ⊢ ( 𝑤 = 𝑊 → { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } = { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 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 ) = ( 𝑅 ‘ 𝑓 ) )
17 16 breq1d ⊢ ( 𝑤 = 𝑊 → ( ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 ↔ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 ) )
18 13 17 rabeqbidv ⊢ ( 𝑤 = 𝑊 → { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } )
19 11 18 mpteq12dv ⊢ ( 𝑤 = 𝑊 → ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) )
20 eqid ⊢ ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) )
21 1 fvexi ⊢ 𝐵 ∈ V
22 21 mptrabex ⊢ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) ∈ V
23 19 20 22 fvmpt ⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤 } ↦ { 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ∣ ( ( ( trL ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑓 ) ≤ 𝑥 } ) ) ‘ 𝑊 ) = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) )
24 9 23 sylan9eq ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ { 𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊 } ↦ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑥 } ) )