Theorem erngfset-rN 34419

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 3066	. 2 |- ( K e. V -> K e. _V )
2		fveq2 5888	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		erngset.h-r	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2514	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 5888	. . . . . . 7 |- ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
6	5	fveq1d 5890	. . . . . 6 |- ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
7	6	opeq2d 4187	. . . . 5 |- ( k = K -> <. ( Base ` ndx ) , ( ( TEndo ` k ) ` w ) >. = <. ( Base ` ndx ) , ( ( TEndo ` K ) ` w ) >. )
8		fveq2 5888	. . . . . . . . 9 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
9	8	fveq1d 5890	. . . . . . . 8 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
10	9	mpteq1d 4498	. . . . . . 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 6380	. . . . . 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 4187	. . . . 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 2463	. . . . . . 7 |- ( k = K -> ( t o. s ) = ( t o. s ) )
14	6, 6, 13	mpt2eq123dv 6380	. . . . . 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 4187	. . . . 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 4079	. . . 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 4495	. . 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 34368	. . 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 5898	. . . . 5 |- ( LHyp ` K ) e. _V
20	3, 19	eqeltri 2536	. . . 4 |- H e. _V
21	20	mptex 6161	. . 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 5971	. 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 1455   e. wcel 1898   _Vcvv 3057   {ctp 3984   <.cop 3986   |-> cmpt 4475   o. ccom 4857   ` cfv 5601   |-> cmpt2 6317   ndxcnx 15167   Basecbs 15170   +g cplusg 15239   .rcmulr 15240   LHypclh 33594   LTrncltrn 33711   TEndoctendo 34364   EDRingRcedring-rN 34366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-oprab 6319  df-mpt2 6320  df-edring-rN 34368

This theorem is referenced by:  erngset-rN  34420