Theorem dvafset 34617

Description: The constructed partial vector space A for a lattice K. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Hypothesis

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

Assertion

Ref	Expression
dvafset	|- ( K e. V -> ( DVecA ` K ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) )

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

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

Proof of Theorem dvafset

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 5892	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		dvaset.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2514	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 5892	. . . . . . . 8 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6	5	fveq1d 5894	. . . . . . 7 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7	6	opeq2d 4187	. . . . . 6 |- ( k = K -> <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. = <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. )
8		eqidd 2463	. . . . . . . 8 |- ( k = K -> ( f o. g ) = ( f o. g ) )
9	6, 6, 8	mpt2eq123dv 6385	. . . . . . 7 |- ( k = K -> ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) = ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) )
10	9	opeq2d 4187	. . . . . 6 |- ( k = K -> <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. = <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. )
11		fveq2 5892	. . . . . . . 8 |- ( k = K -> ( EDRing ` k ) = ( EDRing ` K ) )
12	11	fveq1d 5894	. . . . . . 7 |- ( k = K -> ( ( EDRing ` k ) ` w ) = ( ( EDRing ` K ) ` w ) )
13	12	opeq2d 4187	. . . . . 6 |- ( k = K -> <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. = <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. )
14	7, 10, 13	tpeq123d 4079	. . . . 5 |- ( k = K -> { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } = { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } )
15		fveq2 5892	. . . . . . . . 9 |- ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
16	15	fveq1d 5894	. . . . . . . 8 |- ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
17		eqidd 2463	. . . . . . . 8 |- ( k = K -> ( s ` f ) = ( s ` f ) )
18	16, 6, 17	mpt2eq123dv 6385	. . . . . . 7 |- ( k = K -> ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( LTrn ` k ) ` w ) |-> ( s ` f ) ) = ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) )
19	18	opeq2d 4187	. . . . . 6 |- ( k = K -> <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( LTrn ` k ) ` w ) |-> ( s ` f ) ) >. = <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. )
20	19	sneqd 3992	. . . . 5 |- ( k = K -> { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( LTrn ` k ) ` w ) |-> ( s ` f ) ) >. } = { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } )
21	14, 20	uneq12d 3601	. . . 4 |- ( k = K -> ( { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( LTrn ` k ) ` w ) |-> ( s ` f ) ) >. } ) = ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) )
22	4, 21	mpteq12dv 4497	. . 3 |- ( k = K -> ( w e. ( LHyp ` k ) |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( LTrn ` k ) ` w ) |-> ( s ` f ) ) >. } ) ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) )
23		df-dveca 34616	. . 3 |- DVecA = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` k ) ` w ) , g e. ( ( LTrn ` k ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` k ) ` w ) , f e. ( ( LTrn ` k ) ` w ) |-> ( s ` f ) ) >. } ) ) )
24		fvex 5902	. . . . 5 |- ( LHyp ` K ) e. _V
25	3, 24	eqeltri 2536	. . . 4 |- H e. _V
26	25	mptex 6166	. . 3 |- ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) e. _V
27	22, 23, 26	fvmpt 5976	. 2 |- ( K e. _V -> ( DVecA ` K ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) )
28	1, 27	syl 17	1 |- ( K e. V -> ( DVecA ` K ) = ( w e. H |-> ( { <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. , <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) ) )

Syntax hints:   -> wi 4   = wceq 1455   e. wcel 1898   _Vcvv 3057   u. cun 3414   {csn 3980   {ctp 3984   <.cop 3986   |-> cmpt 4477   o. ccom 4860   ` cfv 5605   |-> cmpt2 6322   ndxcnx 15173   Basecbs 15176   +g cplusg 15245  Scalarcsca 15248   .scvsca 15249   LHypclh 33595   LTrncltrn 33712   TEndoctendo 34365   EDRingcedring 34366   DVecAcdveca 34615

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 4531  ax-sep 4541  ax-nul 4550  ax-pr 4656

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 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-oprab 6324  df-mpt2 6325  df-dveca 34616

This theorem is referenced by:  dvaset  34618