Theorem dvaset 34649

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

Hypotheses

Ref	Expression
dvaset.h	|- H = ( LHyp ` K )
dvaset.t	|- T = ( ( LTrn ` K ) ` W )
dvaset.e	|- E = ( ( TEndo ` K ) ` W )
dvaset.d	|- D = ( ( EDRing ` K ) ` W )
dvaset.u	|- U = ( ( DVecA ` K ) ` W )

Assertion

Ref	Expression
dvaset	|- ( ( K e. X /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) )

Distinct variable groups:   f, g, s, K   f, W, g, s

Allowed substitution hints:   D( f, g, s)   T( f, g, s)   U( f, g, s)   E( f, g, s)   H( f, g, s)   X( f, g, s)

Proof of Theorem dvaset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvaset.u	. 2 |- U = ( ( DVecA ` K ) ` W )
2		dvaset.h	. . . . 5 |- H = ( LHyp ` K )
3	2	dvafset 34648	. . . 4 |- ( K e. X -> ( 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 ) ) >. } ) ) )
4	3	fveq1d 5693	. . 3 |- ( K e. X -> ( ( DVecA ` K ) ` W ) = ( ( 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 ) ) >. } ) ) ` W ) )
5		fveq2 5691	. . . . . . . 8 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
6		dvaset.t	. . . . . . . 8 |- T = ( ( LTrn ` K ) ` W )
7	5, 6	syl6eqr 2493	. . . . . . 7 |- ( w = W -> ( ( LTrn ` K ) ` w ) = T )
8	7	opeq2d 4066	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. = <. ( Base ` ndx ) , T >. )
9		eqidd 2444	. . . . . . . 8 |- ( w = W -> ( f o. g ) = ( f o. g ) )
10	7, 7, 9	mpt2eq123dv 6148	. . . . . . 7 |- ( w = W -> ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) = ( f e. T , g e. T |-> ( f o. g ) ) )
11	10	opeq2d 4066	. . . . . 6 |- ( w = W -> <. ( +g ` ndx ) , ( f e. ( ( LTrn ` K ) ` w ) , g e. ( ( LTrn ` K ) ` w ) |-> ( f o. g ) ) >. = <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. )
12		fveq2 5691	. . . . . . . 8 |- ( w = W -> ( ( EDRing ` K ) ` w ) = ( ( EDRing ` K ) ` W ) )
13		dvaset.d	. . . . . . . 8 |- D = ( ( EDRing ` K ) ` W )
14	12, 13	syl6eqr 2493	. . . . . . 7 |- ( w = W -> ( ( EDRing ` K ) ` w ) = D )
15	14	opeq2d 4066	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. = <. (Scalar ` ndx ) , D >. )
16	8, 11, 15	tpeq123d 3969	. . . . 5 |- ( w = 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 ) >. } = { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } )
17		fveq2 5691	. . . . . . . . 9 |- ( w = W -> ( ( TEndo ` K ) ` w ) = ( ( TEndo ` K ) ` W ) )
18		dvaset.e	. . . . . . . . 9 |- E = ( ( TEndo ` K ) ` W )
19	17, 18	syl6eqr 2493	. . . . . . . 8 |- ( w = W -> ( ( TEndo ` K ) ` w ) = E )
20		eqidd 2444	. . . . . . . 8 |- ( w = W -> ( s ` f ) = ( s ` f ) )
21	19, 7, 20	mpt2eq123dv 6148	. . . . . . 7 |- ( w = W -> ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) = ( s e. E , f e. T |-> ( s ` f ) ) )
22	21	opeq2d 4066	. . . . . 6 |- ( w = W -> <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. = <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. )
23	22	sneqd 3889	. . . . 5 |- ( w = W -> { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } = { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } )
24	16, 23	uneq12d 3511	. . . 4 |- ( w = 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 ) >. } u. { <. ( .s ` ndx ) , ( s e. ( ( TEndo ` K ) ` w ) , f e. ( ( LTrn ` K ) ` w ) |-> ( s ` f ) ) >. } ) = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) )
25		eqid 2443	. . . 4 |- ( 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 ) ) >. } ) ) = ( 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 ) ) >. } ) )
26		tpex 6379	. . . . 5 |- { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } e. _V
27		snex 4533	. . . . 5 |- { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } e. _V
28	26, 27	unex 6378	. . . 4 |- ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) e. _V
29	24, 25, 28	fvmpt 5774	. . 3 |- ( W e. H -> ( ( 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 ) ) >. } ) ) ` W ) = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) )
30	4, 29	sylan9eq 2495	. 2 |- ( ( K e. X /\ W e. H ) -> ( ( DVecA ` K ) ` W ) = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) )
31	1, 30	syl5eq 2487	1 |- ( ( K e. X /\ W e. H ) -> U = ( { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } u. { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   u. cun 3326   {csn 3877   {ctp 3881   <.cop 3883   e. cmpt 4350   o. ccom 4844   ` cfv 5418   e. cmpt2 6093   ndxcnx 14171   Basecbs 14174   +g cplusg 14238  Scalarcsca 14241   .scvsca 14242   LHypclh 33628   LTrncltrn 33745   TEndoctendo 34396   EDRingcedring 34397   DVecAcdveca 34646

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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-un 6372

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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-oprab 6095  df-mpt2 6096  df-dveca 34647

This theorem is referenced by:  dvasca  34650  dvavbase  34657  dvafvadd  34658  dvafvsca  34660  dvaabl  34669