Theorem dvafset 34541

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 3089	. 2 |- ( K e. V -> K e. _V )
2		fveq2 5882	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		dvaset.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2481	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 5882	. . . . . . . 8 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6	5	fveq1d 5884	. . . . . . 7 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7	6	opeq2d 4194	. . . . . 6 |- ( k = K -> <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. = <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. )
8		eqidd 2423	. . . . . . . 8 |- ( k = K -> ( f o. g ) = ( f o. g ) )
9	6, 6, 8	mpt2eq123dv 6368	. . . . . . 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 4194	. . . . . 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 5882	. . . . . . . 8 |- ( k = K -> ( EDRing ` k ) = ( EDRing ` K ) )
12	11	fveq1d 5884	. . . . . . 7 |- ( k = K -> ( ( EDRing ` k ) ` w ) = ( ( EDRing ` K ) ` w ) )
13	12	opeq2d 4194	. . . . . 6 |- ( k = K -> <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. = <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. )
14	7, 10, 13	tpeq123d 4094	. . . . 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 5882	. . . . . . . . 9 |- ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
16	15	fveq1d 5884	. . . . . . . 8 |- ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
17		eqidd 2423	. . . . . . . 8 |- ( k = K -> ( s ` f ) = ( s ` f ) )
18	16, 6, 17	mpt2eq123dv 6368	. . . . . . 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 4194	. . . . . 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 4010	. . . . 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 3621	. . . 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 4502	. . 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 34540	. . 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 5892	. . . . 5 |- ( LHyp ` K ) e. _V
25	3, 24	eqeltri 2503	. . . 4 |- H e. _V
26	25	mptex 6152	. . 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 5965	. 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 1437   e. wcel 1872   _Vcvv 3080   u. cun 3434   {csn 3998   {ctp 4002   <.cop 4004   |-> cmpt 4482   o. ccom 4857   ` cfv 5601   |-> cmpt2 6308   ndxcnx 15118   Basecbs 15121   +g cplusg 15190  Scalarcsca 15193   .scvsca 15194   LHypclh 33519   LTrncltrn 33636   TEndoctendo 34289   EDRingcedring 34290   DVecAcdveca 34539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pr 4660

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  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 6310  df-mpt2 6311  df-dveca 34540

This theorem is referenced by:  dvaset  34542