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Metamath Proof Explorer

Theorem dibfval

Description: The partial isomorphism B for a lattice K . (Contributed by NM, 8-Dec-2013)

Ref Expression
Hypotheses dibval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dibval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dibval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dibval.o ⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
dibval.j ⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
dibval.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion dibfval ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ dom 𝐽 ↦ ( ( 𝐽 ‘ 𝑥 ) × { 0 } ) ) )

Proof

Step Hyp Ref Expression
1 dibval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dibval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 dibval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 dibval.o ⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
5 dibval.j ⊢ 𝐽 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
6 dibval.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
7 1 2 dibffval ⊢ ( 𝐾 ∈ 𝑉 → ( DIsoB ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) )
8 7 fveq1d ⊢ ( 𝐾 ∈ 𝑉 → ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) ‘ 𝑊 ) )
9 6 8 eqtrid ⊢ ( 𝐾 ∈ 𝑉 → 𝐼 = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) ‘ 𝑊 ) )
10 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) )
11 10 5 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) = 𝐽 )
12 11 dmeqd ⊢ ( 𝑤 = 𝑊 → dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) = dom 𝐽 )
13 11 fveq1d ⊢ ( 𝑤 = 𝑊 → ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) = ( 𝐽 ‘ 𝑥 ) )
14 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
15 14 3 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝑇 )
16 eqidd ⊢ ( 𝑤 = 𝑊 → ( I ↾ 𝐵 ) = ( I ↾ 𝐵 ) )
17 15 16 mpteq12dv ⊢ ( 𝑤 = 𝑊 → ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) ) )
18 17 4 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) = 0 )
19 18 sneqd ⊢ ( 𝑤 = 𝑊 → { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } = { 0 } )
20 13 19 xpeq12d ⊢ ( 𝑤 = 𝑊 → ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) = ( ( 𝐽 ‘ 𝑥 ) × { 0 } ) )
21 12 20 mpteq12dv ⊢ ( 𝑤 = 𝑊 → ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) = ( 𝑥 ∈ dom 𝐽 ↦ ( ( 𝐽 ‘ 𝑥 ) × { 0 } ) ) )
22 eqid ⊢ ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) )
23 5 fvexi ⊢ 𝐽 ∈ V
24 23 dmex ⊢ dom 𝐽 ∈ V
25 24 mptex ⊢ ( 𝑥 ∈ dom 𝐽 ↦ ( ( 𝐽 ‘ 𝑥 ) × { 0 } ) ) ∈ V
26 21 22 25 fvmpt ⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑥 ∈ dom ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑤 ) ‘ 𝑥 ) × { ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( I ↾ 𝐵 ) ) } ) ) ) ‘ 𝑊 ) = ( 𝑥 ∈ dom 𝐽 ↦ ( ( 𝐽 ‘ 𝑥 ) × { 0 } ) ) )
27 9 26 sylan9eq ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ dom 𝐽 ↦ ( ( 𝐽 ‘ 𝑥 ) × { 0 } ) ) )