Theorem dvaset 34618

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 34617	. . . 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 5894	. . 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 5892	. . . . . . . 8 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
6		dvaset.t	. . . . . . . 8 |- T = ( ( LTrn ` K ) ` W )
7	5, 6	syl6eqr 2514	. . . . . . 7 |- ( w = W -> ( ( LTrn ` K ) ` w ) = T )
8	7	opeq2d 4187	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. = <. ( Base ` ndx ) , T >. )
9		eqidd 2463	. . . . . . . 8 |- ( w = W -> ( f o. g ) = ( f o. g ) )
10	7, 7, 9	mpt2eq123dv 6385	. . . . . . 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 4187	. . . . . 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 5892	. . . . . . . 8 |- ( w = W -> ( ( EDRing ` K ) ` w ) = ( ( EDRing ` K ) ` W ) )
13		dvaset.d	. . . . . . . 8 |- D = ( ( EDRing ` K ) ` W )
14	12, 13	syl6eqr 2514	. . . . . . 7 |- ( w = W -> ( ( EDRing ` K ) ` w ) = D )
15	14	opeq2d 4187	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. = <. (Scalar ` ndx ) , D >. )
16	8, 11, 15	tpeq123d 4079	. . . . 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 5892	. . . . . . . . 9 |- ( w = W -> ( ( TEndo ` K ) ` w ) = ( ( TEndo ` K ) ` W ) )
18		dvaset.e	. . . . . . . . 9 |- E = ( ( TEndo ` K ) ` W )
19	17, 18	syl6eqr 2514	. . . . . . . 8 |- ( w = W -> ( ( TEndo ` K ) ` w ) = E )
20		eqidd 2463	. . . . . . . 8 |- ( w = W -> ( s ` f ) = ( s ` f ) )
21	19, 7, 20	mpt2eq123dv 6385	. . . . . . 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 4187	. . . . . 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 3992	. . . . 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 3601	. . . 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 2462	. . . 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 6622	. . . . 5 |- { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } e. _V
27		snex 4658	. . . . 5 |- { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } e. _V
28	26, 27	unex 6621	. . . 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 5976	. . 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 2516	. 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 2508	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 375   = wceq 1455   e. wcel 1898   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-8 1900  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  ax-un 6615

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:  dvasca  34619  dvavbase  34626  dvafvadd  34627  dvafvsca  34629  dvaabl  34638