Theorem dvafset 34646

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 2980	. 2 |- ( K e. V -> K e. _V )
2		fveq2 5690	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		dvaset.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2492	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 5690	. . . . . . . 8 |- ( k = K -> ( LTrn ` k ) = ( LTrn ` K ) )
6	5	fveq1d 5692	. . . . . . 7 |- ( k = K -> ( ( LTrn ` k ) ` w ) = ( ( LTrn ` K ) ` w ) )
7	6	opeq2d 4065	. . . . . 6 |- ( k = K -> <. ( Base ` ndx ) , ( ( LTrn ` k ) ` w ) >. = <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. )
8		eqidd 2443	. . . . . . . 8 |- ( k = K -> ( f o. g ) = ( f o. g ) )
9	6, 6, 8	mpt2eq123dv 6147	. . . . . . 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 4065	. . . . . 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 5690	. . . . . . . 8 |- ( k = K -> ( EDRing ` k ) = ( EDRing ` K ) )
12	11	fveq1d 5692	. . . . . . 7 |- ( k = K -> ( ( EDRing ` k ) ` w ) = ( ( EDRing ` K ) ` w ) )
13	12	opeq2d 4065	. . . . . 6 |- ( k = K -> <. (Scalar ` ndx ) , ( ( EDRing ` k ) ` w ) >. = <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. )
14	7, 10, 13	tpeq123d 3968	. . . . 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 5690	. . . . . . . . 9 |- ( k = K -> ( TEndo ` k ) = ( TEndo ` K ) )
16	15	fveq1d 5692	. . . . . . . 8 |- ( k = K -> ( ( TEndo ` k ) ` w ) = ( ( TEndo ` K ) ` w ) )
17		eqidd 2443	. . . . . . . 8 |- ( k = K -> ( s ` f ) = ( s ` f ) )
18	16, 6, 17	mpt2eq123dv 6147	. . . . . . 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 4065	. . . . . 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 3888	. . . . 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 3510	. . . 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 4369	. . 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 34645	. . 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 5700	. . . . 5 |- ( LHyp ` K ) e. _V
25	3, 24	eqeltri 2512	. . . 4 |- H e. _V
26	25	mptex 5947	. . 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 5773	. 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 16	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 1369   e. wcel 1756   _Vcvv 2971   u. cun 3325   {csn 3876   {ctp 3880   <.cop 3882   e. cmpt 4349   o. ccom 4843   ` cfv 5417   e. cmpt2 6092   ndxcnx 14170   Basecbs 14173   +g cplusg 14237  Scalarcsca 14240   .scvsca 14241   LHypclh 33626   LTrncltrn 33743   TEndoctendo 34394   EDRingcedring 34395   DVecAcdveca 34644

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 4402  ax-sep 4412  ax-nul 4420  ax-pr 4530

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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-oprab 6094  df-mpt2 6095  df-dveca 34645

This theorem is referenced by:  dvaset  34647