Theorem ldualset 35266

Description: Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the opp_r ring, for convenience. (Contributed by NM, 18-Oct-2014.)

Hypotheses

Ref	Expression
ldualset.v	|- V = ( Base ` W )
ldualset.a	|- .+ = ( +g ` R )
ldualset.p	|- .+b = ( oF .+ |` ( F X. F ) )
ldualset.f	|- F = (LFnl ` W )
ldualset.d	|- D = (LDual ` W )
ldualset.r	|- R = (Scalar ` W )
ldualset.k	|- K = ( Base ` R )
ldualset.t	|- .x. = ( .r ` R )
ldualset.o	|- O = (oppr ` R )
ldualset.s	|- .xb = ( k e. K , f e. F |-> ( f oF .x. ( V X. { k } ) ) )
ldualset.w	|- ( ph -> W e. X )

Assertion

Ref	Expression
ldualset	|- ( ph -> D = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )

Distinct variable group:   f, k, W

Allowed substitution hints:   ph( f, k)   D( f, k)   .+ ( f, k)   .+b ( f, k)   R( f, k)   .xb ( f, k)   .x. ( f, k)   F( f, k)   K( f, k)   O( f, k)   V( f, k)   X( f, k)

Proof of Theorem ldualset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ldualset.w	. 2 |- ( ph -> W e. X )
2		elex 3115	. 2 |- ( W e. X -> W e. _V )
3		ldualset.d	. . 3 |- D = (LDual ` W )
4		fveq2 5848	. . . . . . . 8 |- ( w = W -> (LFnl ` w ) = (LFnl ` W ) )
5		ldualset.f	. . . . . . . 8 |- F = (LFnl ` W )
6	4, 5	syl6eqr 2513	. . . . . . 7 |- ( w = W -> (LFnl ` w ) = F )
7	6	opeq2d 4210	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , (LFnl ` w ) >. = <. ( Base ` ndx ) , F >. )
8		fveq2 5848	. . . . . . . . . . . . 13 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
9		ldualset.r	. . . . . . . . . . . . 13 |- R = (Scalar ` W )
10	8, 9	syl6eqr 2513	. . . . . . . . . . . 12 |- ( w = W -> (Scalar ` w ) = R )
11	10	fveq2d 5852	. . . . . . . . . . 11 |- ( w = W -> ( +g ` (Scalar ` w ) ) = ( +g ` R ) )
12		ldualset.a	. . . . . . . . . . 11 |- .+ = ( +g ` R )
13	11, 12	syl6eqr 2513	. . . . . . . . . 10 |- ( w = W -> ( +g ` (Scalar ` w ) ) = .+ )
14		ofeq 6515	. . . . . . . . . 10 |- ( ( +g ` (Scalar ` w ) ) = .+ -> oF ( +g ` (Scalar ` w ) ) = oF .+ )
15	13, 14	syl 16	. . . . . . . . 9 |- ( w = W -> oF ( +g ` (Scalar ` w ) ) = oF .+ )
16	6	sqxpeqd 5014	. . . . . . . . 9 |- ( w = W -> ( (LFnl ` w ) X. (LFnl ` w ) ) = ( F X. F ) )
17	15, 16	reseq12d 5263	. . . . . . . 8 |- ( w = W -> ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) = ( oF .+ |` ( F X. F ) ) )
18		ldualset.p	. . . . . . . 8 |- .+b = ( oF .+ |` ( F X. F ) )
19	17, 18	syl6eqr 2513	. . . . . . 7 |- ( w = W -> ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) = .+b )
20	19	opeq2d 4210	. . . . . 6 |- ( w = W -> <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) >. = <. ( +g ` ndx ) , .+b >. )
21	10	fveq2d 5852	. . . . . . . 8 |- ( w = W -> (oppr ` (Scalar ` w ) ) = (oppr ` R ) )
22		ldualset.o	. . . . . . . 8 |- O = (oppr ` R )
23	21, 22	syl6eqr 2513	. . . . . . 7 |- ( w = W -> (oppr ` (Scalar ` w ) ) = O )
24	23	opeq2d 4210	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , (oppr ` (Scalar ` w ) ) >. = <. (Scalar ` ndx ) , O >. )
25	7, 20, 24	tpeq123d 4110	. . . . 5 |- ( w = W -> { <. ( Base ` ndx ) , (LFnl ` w ) >. , <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) >. , <. (Scalar ` ndx ) , (oppr ` (Scalar ` w ) ) >. } = { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } )
26	10	fveq2d 5852	. . . . . . . . . 10 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` R ) )
27		ldualset.k	. . . . . . . . . 10 |- K = ( Base ` R )
28	26, 27	syl6eqr 2513	. . . . . . . . 9 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
29	10	fveq2d 5852	. . . . . . . . . . . 12 |- ( w = W -> ( .r ` (Scalar ` w ) ) = ( .r ` R ) )
30		ldualset.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
31	29, 30	syl6eqr 2513	. . . . . . . . . . 11 |- ( w = W -> ( .r ` (Scalar ` w ) ) = .x. )
32		ofeq 6515	. . . . . . . . . . 11 |- ( ( .r ` (Scalar ` w ) ) = .x. -> oF ( .r ` (Scalar ` w ) ) = oF .x. )
33	31, 32	syl 16	. . . . . . . . . 10 |- ( w = W -> oF ( .r ` (Scalar ` w ) ) = oF .x. )
34		eqidd 2455	. . . . . . . . . 10 |- ( w = W -> f = f )
35		fveq2 5848	. . . . . . . . . . . 12 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
36		ldualset.v	. . . . . . . . . . . 12 |- V = ( Base ` W )
37	35, 36	syl6eqr 2513	. . . . . . . . . . 11 |- ( w = W -> ( Base ` w ) = V )
38	37	xpeq1d 5011	. . . . . . . . . 10 |- ( w = W -> ( ( Base ` w ) X. { k } ) = ( V X. { k } ) )
39	33, 34, 38	oveq123d 6291	. . . . . . . . 9 |- ( w = W -> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) = ( f oF .x. ( V X. { k } ) ) )
40	28, 6, 39	mpt2eq123dv 6332	. . . . . . . 8 |- ( w = W -> ( k e. ( Base ` (Scalar ` w ) ) , f e. (LFnl ` w ) |-> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) ) = ( k e. K , f e. F |-> ( f oF .x. ( V X. { k } ) ) ) )
41		ldualset.s	. . . . . . . 8 |- .xb = ( k e. K , f e. F |-> ( f oF .x. ( V X. { k } ) ) )
42	40, 41	syl6eqr 2513	. . . . . . 7 |- ( w = W -> ( k e. ( Base ` (Scalar ` w ) ) , f e. (LFnl ` w ) |-> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) ) = .xb )
43	42	opeq2d 4210	. . . . . 6 |- ( w = W -> <. ( .s ` ndx ) , ( k e. ( Base ` (Scalar ` w ) ) , f e. (LFnl ` w ) |-> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) ) >. = <. ( .s ` ndx ) , .xb >. )
44	43	sneqd 4028	. . . . 5 |- ( w = W -> { <. ( .s ` ndx ) , ( k e. ( Base ` (Scalar ` w ) ) , f e. (LFnl ` w ) |-> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) ) >. } = { <. ( .s ` ndx ) , .xb >. } )
45	25, 44	uneq12d 3645	. . . 4 |- ( w = W -> ( { <. ( Base ` ndx ) , (LFnl ` w ) >. , <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) >. , <. (Scalar ` ndx ) , (oppr ` (Scalar ` w ) ) >. } u. { <. ( .s ` ndx ) , ( k e. ( Base ` (Scalar ` w ) ) , f e. (LFnl ` w ) |-> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) ) >. } ) = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )
46		df-ldual 35265	. . . 4 |- LDual = ( w e. _V |-> ( { <. ( Base ` ndx ) , (LFnl ` w ) >. , <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) >. , <. (Scalar ` ndx ) , (oppr ` (Scalar ` w ) ) >. } u. { <. ( .s ` ndx ) , ( k e. ( Base ` (Scalar ` w ) ) , f e. (LFnl ` w ) |-> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) ) >. } ) )
47		tpex 6572	. . . . 5 |- { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } e. _V
48		snex 4678	. . . . 5 |- { <. ( .s ` ndx ) , .xb >. } e. _V
49	47, 48	unex 6571	. . . 4 |- ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) e. _V
50	45, 46, 49	fvmpt 5931	. . 3 |- ( W e. _V -> (LDual ` W ) = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )
51	3, 50	syl5eq 2507	. 2 |- ( W e. _V -> D = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )
52	1, 2, 51	3syl 20	1 |- ( ph -> D = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 1823   _Vcvv 3106   u. cun 3459   {csn 4016   {ctp 4020   <.cop 4022   X. cxp 4986   |` cres 4990   ` cfv 5570  (class class class)co 6270   |-> cmpt2 6272   oFcof 6511   ndxcnx 14716   Basecbs 14719   +g cplusg 14787   .rcmulr 14788  Scalarcsca 14790   .scvsca 14791  opp_rcoppr 17469  LFnlclfn 35198  LDualcld 35264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-res 5000  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-ldual 35265

This theorem is referenced by:  ldualvbase  35267  ldualfvadd  35269  ldualsca  35273  ldualfvs  35277