Theorem erngfset-rN 36095

Description: The division rings on trace-preserving endomorphisms for a lattice K. (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
erngset.h-r	|- H = ( LHyp ` K )

Assertion

Ref	Expression
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 ) ) >. } ) )

Distinct variable groups:   w, H   f, s, t, w, K

Allowed substitution hints:   H( t, f, s)   V( w, t, f, s)

Proof of Theorem erngfset-rN

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. V -> K e. _V )
2		fveq2 6191	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		erngset.h-r	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . . 7 |- ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
6	5	fveq1d 6193	. . . . . 6 |- ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
7	6	opeq2d 4409	. . . . 5 |- ( k = K -> <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. )
8		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
9	8	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
10	9	mpteq1d 4738	. . . . . . 7 |- ( k = K -> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) = ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) )
11	6, 6, 10	mpt2eq123dv 6717	. . . . . 6 |- ( k = K -> ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( f e. ( ( LTrn ` k ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) = ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) )
12	11	opeq2d 4409	. . . . 5 |- ( k = K -> <. ( +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. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( ( s ` f ) o. ( t ` f ) ) ) ) >. )
13		eqidd 2623	. . . . . . 7 |- ( k = K -> ( t o. s ) = ( t o. s ) )
14	6, 6, 13	mpt2eq123dv 6717	. . . . . 6 |- ( k = K -> ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) = ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) )
15	14	opeq2d 4409	. . . . 5 |- ( k = K -> <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( t o. s ) ) >. = <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( t o. s ) ) >. )
16	7, 12, 15	tpeq123d 4283	. . . 4 |- ( k = K -> { <. ( 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 ) ) >. } )
17	4, 16	mpteq12dv 4733	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> { <. ( 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 ) ) >. } ) )
18		df-edring-rN 36044	. . 3 |- EDRingR = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { <. ( 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 ) ) >. } ) )
19		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
20	3, 19	eqeltri 2697	. . . 4 |- H e. _V
21	20	mptex 6486	. . 3 |- ( 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 ) ) >. } ) e. _V
22	17, 18, 21	fvmpt 6282	. 2 |- ( 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 ) ) >. } ) )
23	1, 22	syl 17	1 |- ( 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 ) ) >. } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   {ctp 4181   <.cop 4183   |-> cmpt 4729   o. ccom 5118   ` cfv 5888   |-> cmpt2 6652   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   EDRingRcedring-rN 36042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-edring-rN 36044

This theorem is referenced by:  erngset-rN  36096