Theorem ldualset 32603

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 3031	. 2 |- ( W e. X -> W e. _V )
3		ldualset.d	. . 3 |- D = (LDual ` W )
4		fveq2 5825	. . . . . . . 8 |- ( w = W -> (LFnl ` w ) = (LFnl ` W ) )
5		ldualset.f	. . . . . . . 8 |- F = (LFnl ` W )
6	4, 5	syl6eqr 2480	. . . . . . 7 |- ( w = W -> (LFnl ` w ) = F )
7	6	opeq2d 4137	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , (LFnl ` w ) >. = <. ( Base ` ndx ) , F >. )
8		fveq2 5825	. . . . . . . . . . . . 13 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
9		ldualset.r	. . . . . . . . . . . . 13 |- R = (Scalar ` W )
10	8, 9	syl6eqr 2480	. . . . . . . . . . . 12 |- ( w = W -> (Scalar ` w ) = R )
11	10	fveq2d 5829	. . . . . . . . . . 11 |- ( w = W -> ( +g ` (Scalar ` w ) ) = ( +g ` R ) )
12		ldualset.a	. . . . . . . . . . 11 |- .+ = ( +g ` R )
13	11, 12	syl6eqr 2480	. . . . . . . . . 10 |- ( w = W -> ( +g ` (Scalar ` w ) ) = .+ )
14		ofeq 6491	. . . . . . . . . 10 |- ( ( +g ` (Scalar ` w ) ) = .+ -> oF ( +g ` (Scalar ` w ) ) = oF .+ )
15	13, 14	syl 17	. . . . . . . . 9 |- ( w = W -> oF ( +g ` (Scalar ` w ) ) = oF .+ )
16	6	sqxpeqd 4822	. . . . . . . . 9 |- ( w = W -> ( (LFnl ` w ) X. (LFnl ` w ) ) = ( F X. F ) )
17	15, 16	reseq12d 5068	. . . . . . . 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 2480	. . . . . . 7 |- ( w = W -> ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) = .+b )
20	19	opeq2d 4137	. . . . . 6 |- ( w = W -> <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) >. = <. ( +g ` ndx ) , .+b >. )
21	10	fveq2d 5829	. . . . . . . 8 |- ( w = W -> (oppr ` (Scalar ` w ) ) = (oppr ` R ) )
22		ldualset.o	. . . . . . . 8 |- O = (oppr ` R )
23	21, 22	syl6eqr 2480	. . . . . . 7 |- ( w = W -> (oppr ` (Scalar ` w ) ) = O )
24	23	opeq2d 4137	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , (oppr ` (Scalar ` w ) ) >. = <. (Scalar ` ndx ) , O >. )
25	7, 20, 24	tpeq123d 4037	. . . . 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 5829	. . . . . . . . . 10 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` R ) )
27		ldualset.k	. . . . . . . . . 10 |- K = ( Base ` R )
28	26, 27	syl6eqr 2480	. . . . . . . . 9 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
29	10	fveq2d 5829	. . . . . . . . . . . 12 |- ( w = W -> ( .r ` (Scalar ` w ) ) = ( .r ` R ) )
30		ldualset.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
31	29, 30	syl6eqr 2480	. . . . . . . . . . 11 |- ( w = W -> ( .r ` (Scalar ` w ) ) = .x. )
32		ofeq 6491	. . . . . . . . . . 11 |- ( ( .r ` (Scalar ` w ) ) = .x. -> oF ( .r ` (Scalar ` w ) ) = oF .x. )
33	31, 32	syl 17	. . . . . . . . . 10 |- ( w = W -> oF ( .r ` (Scalar ` w ) ) = oF .x. )
34		eqidd 2429	. . . . . . . . . 10 |- ( w = W -> f = f )
35		fveq2 5825	. . . . . . . . . . . 12 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
36		ldualset.v	. . . . . . . . . . . 12 |- V = ( Base ` W )
37	35, 36	syl6eqr 2480	. . . . . . . . . . 11 |- ( w = W -> ( Base ` w ) = V )
38	37	xpeq1d 4819	. . . . . . . . . 10 |- ( w = W -> ( ( Base ` w ) X. { k } ) = ( V X. { k } ) )
39	33, 34, 38	oveq123d 6270	. . . . . . . . 9 |- ( w = W -> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) = ( f oF .x. ( V X. { k } ) ) )
40	28, 6, 39	mpt2eq123dv 6311	. . . . . . . 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 2480	. . . . . . 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 4137	. . . . . 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 3953	. . . . 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 3564	. . . 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 32602	. . . 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 6548	. . . . 5 |- { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } e. _V
48		snex 4605	. . . . 5 |- { <. ( .s ` ndx ) , .xb >. } e. _V
49	47, 48	unex 6547	. . . 4 |- ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) e. _V
50	45, 46, 49	fvmpt 5908	. . 3 |- ( W e. _V -> (LDual ` W ) = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )
51	3, 50	syl5eq 2474	. 2 |- ( W e. _V -> D = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )
52	1, 2, 51	3syl 18	1 |- ( ph -> D = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )

Syntax hints:   -> wi 4   = wceq 1437   e. wcel 1872   _Vcvv 3022   u. cun 3377   {csn 3941   {ctp 3945   <.cop 3947   X. cxp 4794   |` cres 4798   ` cfv 5544  (class class class)co 6249   |-> cmpt2 6251   oFcof 6487   ndxcnx 15061   Basecbs 15064   +g cplusg 15133   .rcmulr 15134  Scalarcsca 15136   .scvsca 15137  opp_rcoppr 17793  LFnlclfn 32535  LDualcld 32601

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 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541

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 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-res 4808  df-iota 5508  df-fun 5546  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-ldual 32602

This theorem is referenced by:  ldualvbase  32604  ldualfvadd  32606  ldualsca  32610  ldualfvs  32614