Theorem ldualset 33940

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 3122	. 2 |- ( W e. X -> W e. _V )
3		ldualset.d	. . 3 |- D = (LDual ` W )
4		fveq2 5866	. . . . . . . 8 |- ( w = W -> (LFnl ` w ) = (LFnl ` W ) )
5		ldualset.f	. . . . . . . 8 |- F = (LFnl ` W )
6	4, 5	syl6eqr 2526	. . . . . . 7 |- ( w = W -> (LFnl ` w ) = F )
7	6	opeq2d 4220	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , (LFnl ` w ) >. = <. ( Base ` ndx ) , F >. )
8		fveq2 5866	. . . . . . . . . . . . 13 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
9		ldualset.r	. . . . . . . . . . . . 13 |- R = (Scalar ` W )
10	8, 9	syl6eqr 2526	. . . . . . . . . . . 12 |- ( w = W -> (Scalar ` w ) = R )
11	10	fveq2d 5870	. . . . . . . . . . 11 |- ( w = W -> ( +g ` (Scalar ` w ) ) = ( +g ` R ) )
12		ldualset.a	. . . . . . . . . . 11 |- .+ = ( +g ` R )
13	11, 12	syl6eqr 2526	. . . . . . . . . 10 |- ( w = W -> ( +g ` (Scalar ` w ) ) = .+ )
14		ofeq 6526	. . . . . . . . . 10 |- ( ( +g ` (Scalar ` w ) ) = .+ -> oF ( +g ` (Scalar ` w ) ) = oF .+ )
15	13, 14	syl 16	. . . . . . . . 9 |- ( w = W -> oF ( +g ` (Scalar ` w ) ) = oF .+ )
16	6, 6	xpeq12d 5024	. . . . . . . . 9 |- ( w = W -> ( (LFnl ` w ) X. (LFnl ` w ) ) = ( F X. F ) )
17	15, 16	reseq12d 5274	. . . . . . . 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 2526	. . . . . . 7 |- ( w = W -> ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) = .+b )
20	19	opeq2d 4220	. . . . . 6 |- ( w = W -> <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) >. = <. ( +g ` ndx ) , .+b >. )
21	10	fveq2d 5870	. . . . . . . 8 |- ( w = W -> (oppr ` (Scalar ` w ) ) = (oppr ` R ) )
22		ldualset.o	. . . . . . . 8 |- O = (oppr ` R )
23	21, 22	syl6eqr 2526	. . . . . . 7 |- ( w = W -> (oppr ` (Scalar ` w ) ) = O )
24	23	opeq2d 4220	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , (oppr ` (Scalar ` w ) ) >. = <. (Scalar ` ndx ) , O >. )
25	7, 20, 24	tpeq123d 4121	. . . . 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 5870	. . . . . . . . . 10 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` R ) )
27		ldualset.k	. . . . . . . . . 10 |- K = ( Base ` R )
28	26, 27	syl6eqr 2526	. . . . . . . . 9 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
29	10	fveq2d 5870	. . . . . . . . . . . 12 |- ( w = W -> ( .r ` (Scalar ` w ) ) = ( .r ` R ) )
30		ldualset.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
31	29, 30	syl6eqr 2526	. . . . . . . . . . 11 |- ( w = W -> ( .r ` (Scalar ` w ) ) = .x. )
32		ofeq 6526	. . . . . . . . . . 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 2468	. . . . . . . . . 10 |- ( w = W -> f = f )
35		fveq2 5866	. . . . . . . . . . . 12 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
36		ldualset.v	. . . . . . . . . . . 12 |- V = ( Base ` W )
37	35, 36	syl6eqr 2526	. . . . . . . . . . 11 |- ( w = W -> ( Base ` w ) = V )
38	37	xpeq1d 5022	. . . . . . . . . 10 |- ( w = W -> ( ( Base ` w ) X. { k } ) = ( V X. { k } ) )
39	33, 34, 38	oveq123d 6305	. . . . . . . . 9 |- ( w = W -> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) = ( f oF .x. ( V X. { k } ) ) )
40	28, 6, 39	mpt2eq123dv 6343	. . . . . . . 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 2526	. . . . . . 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 4220	. . . . . 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 4039	. . . . 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 3659	. . . 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 33939	. . . 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 6583	. . . . 5 |- { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } e. _V
48		snex 4688	. . . . 5 |- { <. ( .s ` ndx ) , .xb >. } e. _V
49	47, 48	unex 6582	. . . 4 |- ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) e. _V
50	45, 46, 49	fvmpt 5950	. . 3 |- ( W e. _V -> (LDual ` W ) = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )
51	3, 50	syl5eq 2520	. 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 1379   e. wcel 1767   _Vcvv 3113   u. cun 3474   {csn 4027   {ctp 4031   <.cop 4033   X. cxp 4997   |` cres 5001   ` cfv 5588  (class class class)co 6284   |-> cmpt2 6286   oFcof 6522   ndxcnx 14487   Basecbs 14490   +g cplusg 14555   .rcmulr 14556  Scalarcsca 14558   .scvsca 14559  opp_rcoppr 17072  LFnlclfn 33872  LDualcld 33938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-ldual 33939

This theorem is referenced by:  ldualvbase  33941  ldualfvadd  33943  ldualsca  33947  ldualfvs  33951