Theorem dvhfset 36369

Description: The constructed full vector space H for a lattice K. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Hypothesis

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

Assertion

Ref

Expression

dvhfset

|- ( K e. V -> ( DVecH ` K ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )

Distinct variable groups:   f, g, w, H   f, h, s, K, g, w

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

Proof of Theorem dvhfset

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		dvhset.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6	5	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
8	7	fveq1d 6193	. . . . . . . 8 |- ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
9	6, 8	xpeq12d 5140	. . . . . . 7 |- ( k = K -> ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) = ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) )
10	9	opeq2d 4409	. . . . . 6 |- ( k = K -> <. ( Base ` ndx ) , ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) >. = <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. )
11	6	mpteq1d 4738	. . . . . . . . 9 |- ( k = K -> ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) = ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) )
12	11	opeq2d 4409	. . . . . . . 8 |- ( k = K -> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. = <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. )
13	9, 9, 12	mpt2eq123dv 6717	. . . . . . 7 |- ( k = K -> ( f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) , g e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) = ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) )
14	13	opeq2d 4409	. . . . . 6 |- ( k = K -> <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) , g e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. = <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. )
15		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( EDRing ` k ) = ( EDRing ` K ) )
16	15	fveq1d 6193	. . . . . . 7 |- ( k = K -> ( ( EDRing ` k ) ` w ) = ( ( EDRing ` K ) ` w ) )
17	16	opeq2d 4409	. . . . . 6 |- ( k = K -> <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. = <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. )
18	10, 14, 17	tpeq123d 4283	. . . . 5 |- ( k = K -> { <. ( Base ` ndx ) , ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) , g e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } = { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } )
19		eqidd 2623	. . . . . . . 8 |- ( k = K -> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. = <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. )
20	8, 9, 19	mpt2eq123dv 6717	. . . . . . 7 |- ( k = K -> ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) = ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) )
21	20	opeq2d 4409	. . . . . 6 |- ( k = K -> <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. = <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. )
22	21	sneqd 4189	. . . . 5 |- ( k = K -> { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } = { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } )
23	18, 22	uneq12d 3768	. . . 4 |- ( k = K -> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) , g e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) = ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) )
24	4, 23	mpteq12dv 4733	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) , g e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
25		df-dvech 36368	. . 3 |- DVecH = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) , g e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` k ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( ( LTrn ` k ) ` w ) X. ( ( TEndo ` k ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
26		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
27	3, 26	eqeltri 2697	. . . 4 |- H e. _V
28	27	mptex 6486	. . 3 |- ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) e. _V
29	24, 25, 28	fvmpt 6282	. 2 |- ( K e. _V -> ( DVecH ` K ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )
30	1, 29	syl 17	1 |- ( K e. V -> ( DVecH ` K ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) >. , <. ( +g ` ndx ) , ( f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) , g e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( h e. ( ( LTrn ` K ) ` w ) |-> ( ( ( 2nd ` f ) ` h ) o. ( ( 2nd ` g ) ` h ) ) ) >. ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( ( LTrn ` K ) ` w ) X. ( ( TEndo ` K ) ` w ) ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. ) >. } ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   {ctp 4181   <.cop 4183   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   ` cfv 5888   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ndxcnx 15854   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   EDRingcedring 36041   DVecHcdvh 36367

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-dvech 36368

This theorem is referenced by:  dvhset  36370