Theorem erngfset-rN 34451

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 2981	. 2 |- ( K e. V -> K e. _V )
2		fveq2 5691	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		erngset.h-r	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2493	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 5691	. . . . . . 7 |- ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
6	5	fveq1d 5693	. . . . . 6 |- ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
7	6	opeq2d 4066	. . . . 5 |- ( k = K -> <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. )
8		fveq2 5691	. . . . . . . . 9 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
9	8	fveq1d 5693	. . . . . . . 8 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
10	9	mpteq1d 4373	. . . . . . 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 6148	. . . . . 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 4066	. . . . 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 2444	. . . . . . 7 |- ( k = K -> ( t o. s ) = ( t o. s ) )
14	6, 6, 13	mpt2eq123dv 6148	. . . . . 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 4066	. . . . 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 3969	. . . 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 4370	. . 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 34400	. . 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 5701	. . . . 5 |- ( LHyp ` K ) e. _V
20	3, 19	eqeltri 2513	. . . 4 |- H e. _V
21	20	mptex 5948	. . 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 5774	. 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 16	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 1369   e. wcel 1756   _Vcvv 2972   {ctp 3881   <.cop 3883   e. cmpt 4350   o. ccom 4844   ` cfv 5418   e. cmpt2 6093   ndxcnx 14171   Basecbs 14174   +g cplusg 14238   .rcmulr 14239   LHypclh 33628   LTrncltrn 33745   TEndoctendo 34396   EDRingRcedring-rN 34398

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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531

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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-oprab 6095  df-mpt2 6096  df-edring-rN 34400

This theorem is referenced by:  erngset-rN  34452