Theorem dvaset 36832

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 36831	. . . 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 5874	. . 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 5872	. . . . . . . 8 |- ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
6		dvaset.t	. . . . . . . 8 |- T = ( ( LTrn ` K ) ` W )
7	5, 6	syl6eqr 2516	. . . . . . 7 |- ( w = W -> ( ( LTrn ` K ) ` w ) = T )
8	7	opeq2d 4226	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , ( ( LTrn ` K ) ` w ) >. = <. ( Base ` ndx ) , T >. )
9		eqidd 2458	. . . . . . . 8 |- ( w = W -> ( f o. g ) = ( f o. g ) )
10	7, 7, 9	mpt2eq123dv 6358	. . . . . . 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 4226	. . . . . 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 5872	. . . . . . . 8 |- ( w = W -> ( ( EDRing ` K ) ` w ) = ( ( EDRing ` K ) ` W ) )
13		dvaset.d	. . . . . . . 8 |- D = ( ( EDRing ` K ) ` W )
14	12, 13	syl6eqr 2516	. . . . . . 7 |- ( w = W -> ( ( EDRing ` K ) ` w ) = D )
15	14	opeq2d 4226	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , ( ( EDRing ` K ) ` w ) >. = <. (Scalar ` ndx ) , D >. )
16	8, 11, 15	tpeq123d 4126	. . . . 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 5872	. . . . . . . . 9 |- ( w = W -> ( ( TEndo ` K ) ` w ) = ( ( TEndo ` K ) ` W ) )
18		dvaset.e	. . . . . . . . 9 |- E = ( ( TEndo ` K ) ` W )
19	17, 18	syl6eqr 2516	. . . . . . . 8 |- ( w = W -> ( ( TEndo ` K ) ` w ) = E )
20		eqidd 2458	. . . . . . . 8 |- ( w = W -> ( s ` f ) = ( s ` f ) )
21	19, 7, 20	mpt2eq123dv 6358	. . . . . . 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 4226	. . . . . 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 4044	. . . . 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 3655	. . . 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 2457	. . . 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 6598	. . . . 5 |- { <. ( Base ` ndx ) , T >. , <. ( +g ` ndx ) , ( f e. T , g e. T |-> ( f o. g ) ) >. , <. (Scalar ` ndx ) , D >. } e. _V
27		snex 4697	. . . . 5 |- { <. ( .s ` ndx ) , ( s e. E , f e. T |-> ( s ` f ) ) >. } e. _V
28	26, 27	unex 6597	. . . 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 5956	. . 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 2518	. 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 2510	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 1395   e. wcel 1819   u. cun 3469   {csn 4032   {ctp 4036   <.cop 4038   |-> cmpt 4515   o. ccom 5012   ` cfv 5594   |-> cmpt2 6298   ndxcnx 14640   Basecbs 14643   +g cplusg 14711  Scalarcsca 14714   .scvsca 14715   LHypclh 35809   LTrncltrn 35926   TEndoctendo 36579   EDRingcedring 36580   DVecAcdveca 36829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-dveca 36830

This theorem is referenced by:  dvasca  36833  dvavbase  36840  dvafvadd  36841  dvafvsca  36843  dvaabl  36852