Theorem erngfset 34283

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

Hypothesis

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

Assertion

Ref	Expression
erngfset	|- ( K e. V -> ( EDRing ` 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 ) |-> ( s o. t ) ) >. } ) )

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

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2976	. 2 |- ( K e. V -> K e. _V )
2		fveq2 5686	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		erngset.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2488	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 5686	. . . . . . 7 |- ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
6	5	fveq1d 5688	. . . . . 6 |- ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
7	6	opeq2d 4061	. . . . 5 |- ( k = K -> <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. )
8		fveq2 5686	. . . . . . . . 9 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
9	8	fveq1d 5688	. . . . . . . 8 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
10	9	mpteq1d 4368	. . . . . . 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 6143	. . . . . 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 4061	. . . . 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 2439	. . . . . . 7 |- ( k = K -> ( s o. t ) = ( s o. t ) )
14	6, 6, 13	mpt2eq123dv 6143	. . . . . 6 |- ( k = K -> ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) = ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( s o. t ) ) )
15	14	opeq2d 4061	. . . . 5 |- ( k = K -> <. ( .r ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , t e. ( ( TEndo ` k ) ` w ) |-> ( s o. t ) ) >. = <. ( .r ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , t e. ( ( TEndo ` K ) ` w ) |-> ( s o. t ) ) >. )
16	7, 12, 15	tpeq123d 3964	. . . 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 ) |-> ( s o. t ) ) >. } = { <. ( 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 ) |-> ( s o. t ) ) >. } )
17	4, 16	mpteq12dv 4365	. . 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 ) |-> ( s o. t ) ) >. } ) = ( 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 ) |-> ( s o. t ) ) >. } ) )
18		df-edring 34241	. . 3 |- EDRing = ( 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 ) |-> ( s o. t ) ) >. } ) )
19		fvex 5696	. . . . 5 |- ( LHyp ` K ) e. _V
20	3, 19	eqeltri 2508	. . . 4 |- H e. _V
21	20	mptex 5943	. . 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 ) |-> ( s o. t ) ) >. } ) e. _V
22	17, 18, 21	fvmpt 5769	. 2 |- ( K e. _V -> ( EDRing ` 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 ) |-> ( s o. t ) ) >. } ) )
23	1, 22	syl 16	1 |- ( K e. V -> ( EDRing ` 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 ) |-> ( s o. t ) ) >. } ) )

Syntax hints:   -> wi 4   = wceq 1369   e. wcel 1756   _Vcvv 2967   {ctp 3876   <.cop 3878   e. cmpt 4345   o. ccom 4839   ` cfv 5413   e. cmpt2 6088   ndxcnx 14163   Basecbs 14166   +g cplusg 14230   .rcmulr 14231   LHypclh 33468   LTrncltrn 33585   TEndoctendo 34236   EDRingcedring 34237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-oprab 6090  df-mpt2 6091  df-edring 34241

This theorem is referenced by:  erngset  34284