Metamath Proof Explorer

Theorem erngset-rN

Description: The division ring on trace-preserving endomorphisms for a fiducial co-atom W . (Contributed by NM, 5-Jun-2013) (New usage is discouraged.)

Ref Expression
Hypotheses erngset.h-r 
|- H = ( LHyp ` K )
erngset.t-r 
|- T = ( ( LTrn ` K ) ` W )
erngset.e-r 
|- E = ( ( TEndo ` K ) ` W )
erngset.d-r 
|- D = ( ( EDRingR ` K ) ` W )
Assertion erngset-rN 
|- ( ( K e. V /\ W e. H ) -> D = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )

Proof

Step Hyp Ref Expression
1 erngset.h-r 
 |-  H = ( LHyp ` K )
2 erngset.t-r 
 |-  T = ( ( LTrn ` K ) ` W )
3 erngset.e-r 
 |-  E = ( ( TEndo ` K ) ` W )
4 erngset.d-r 
 |-  D = ( ( EDRingR ` K ) ` W )
5 1 erngfset-rN 
 |-  ( K e. V -> ( EDRingR ` K ) = ( w e. H |-> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } ) )
6 5 fveq1d 
 |-  ( K e. V -> ( ( EDRingR ` K ) ` W ) = ( ( w e. H |-> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } ) ` W ) )
7 4 6 eqtrid 
 |-  ( K e. V -> D = ( ( w e. H |-> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } ) ` W ) )
8 fveq2 
 |-  ( w = W -> ( ( TEndo ` K ) ` w ) = ( ( TEndo ` K ) ` W ) )
9 8 opeq2d 
 |-  ( w = W -> <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` W ) >. )
10 tpeq1 
 |-  ( <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` W ) >. -> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` W ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } )
11 3 opeq2i 
 |-  <. ( Base ` ndx ) , E >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` W ) >.
12 tpeq1 
 |-  ( <. ( Base ` ndx ) , E >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` W ) >. -> { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` W ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } )
13 11 12 ax-mp 
 |-  { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` W ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. }
14 10 13 eqtr4di 
 |-  ( <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` W ) >. -> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } )
15 9 14 syl 
 |-  ( w = W -> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } )
16 8 3 eqtr4di 
 |-  ( w = W -> ( ( TEndo ` K ) ` w ) = E )
17 fveq2 
 |-  ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
18 17 2 eqtr4di 
 |-  ( w = W -> ( ( LTrn ` K ) ` w ) = T )
19 eqidd 
 |-  ( w = W -> ( ( s ` f ) o. ( t ` f ) ) = ( ( s ` f ) o. ( t ` f ) ) )
20 18 19 mpteq12dv 
 |-  ( w = W -> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) = ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
21 16 16 20 mpoeq123dv 
 |-  ( w = W -> ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) )
22 21 opeq2d 
 |-  ( w = W -> <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. = <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. )
23 22 tpeq2d 
 |-  ( w = W -> { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } )
24 eqidd 
 |-  ( w = W -> ( t o. s ) = ( t o. s ) )
25 16 16 24 mpoeq123dv 
 |-  ( w = W -> ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) = ( s e. E , t e. E |-> ( t o. s ) ) )
26 25 opeq2d 
 |-  ( w = W -> <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. = <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. )
27 26 tpeq3d 
 |-  ( w = W -> { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )
28 15 23 27 3eqtrd 
 |-  ( w = W -> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )
29 eqid 
 |-  ( w e. H |-> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } ) = ( w e. H |-> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } )
30 tpex 
 |-  { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } e. _V
31 28 29 30 fvmpt 
 |-  ( W e. H -> ( ( w e. H |-> { <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. , <. ( +g ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. } ) ` W ) = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )
32 7 31 sylan9eq 
 |-  ( ( K e. V /\ W e. H ) -> D = { <. ( Base ` ndx ) , E >. , <. ( +g ` ndx ) , ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. , <. ( .r ` ndx ) , ( s e. E , t e. E |-> ( t o. s ) ) >. } )