Theorem dvaset 34542

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 34541	. . . 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 5884	. . 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 5882	. . . . . . . 8 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
6		dvaset.t	. . . . . . . 8 |- T = ( ( LTrn ` K ) ` W )
7	5, 6	syl6eqr 2481	. . . . . . 7 |- ( w = W -> ( ( LTrn ` K ) ` w ) = T )
8	7	opeq2d 4194	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. = <. ( Base ` ndx ) , T >. )
9		eqidd 2423	. . . . . . . 8 |- ( w = W -> ( f o. g ) = ( f o. g ) )
10	7, 7, 9	mpt2eq123dv 6368	. . . . . . 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 4194	. . . . . 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 5882	. . . . . . . 8 |- ( w = W -> ( ( EDRing ` K ) ` w ) = ( ( EDRing ` K ) ` W ) )
13		dvaset.d	. . . . . . . 8 |- D = ( ( EDRing ` K ) ` W )
14	12, 13	syl6eqr 2481	. . . . . . 7 |- ( w = W -> ( ( EDRing ` K ) ` w ) = D )
15	14	opeq2d 4194	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. = <. (Scalar ` ndx ) , D >. )
16	8, 11, 15	tpeq123d 4094	. . . . 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 5882	. . . . . . . . 9 |- ( w = W -> ( ( TEndo ` K ) ` w ) = ( ( TEndo ` K ) ` W ) )
18		dvaset.e	. . . . . . . . 9 |- E = ( ( TEndo ` K ) ` W )
19	17, 18	syl6eqr 2481	. . . . . . . 8 |- ( w = W -> ( ( TEndo ` K ) ` w ) = E )
20		eqidd 2423	. . . . . . . 8 |- ( w = W -> ( s ` f ) = ( s ` f ) )
21	19, 7, 20	mpt2eq123dv 6368	. . . . . . 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 4194	. . . . . 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 4010	. . . . 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 3621	. . . 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 2422	. . . 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 6605	. . . . 5 |- { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } e. _V
27		snex 4662	. . . . 5 |- { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } e. _V
28	26, 27	unex 6604	. . . 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 5965	. . 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 2483	. 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 2475	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 370   = wceq 1437   e. wcel 1872   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-8 1874  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  ax-un 6598

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:  dvasca  34543  dvavbase  34550  dvafvadd  34551  dvafvsca  34553  dvaabl  34562