Theorem ldualset 34412

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 3212	. 2 |- ( W e. X -> W e. _V )
3		ldualset.d	. . 3 |- D = (LDual ` W )
4		fveq2 6191	. . . . . . . 8 |- ( w = W -> (LFnl ` w ) = (LFnl ` W ) )
5		ldualset.f	. . . . . . . 8 |- F = (LFnl ` W )
6	4, 5	syl6eqr 2674	. . . . . . 7 |- ( w = W -> (LFnl ` w ) = F )
7	6	opeq2d 4409	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , (LFnl ` w ) >. = <. ( Base ` ndx ) , F >. )
8		fveq2 6191	. . . . . . . . . . . . 13 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
9		ldualset.r	. . . . . . . . . . . . 13 |- R = (Scalar ` W )
10	8, 9	syl6eqr 2674	. . . . . . . . . . . 12 |- ( w = W -> (Scalar ` w ) = R )
11	10	fveq2d 6195	. . . . . . . . . . 11 |- ( w = W -> ( +g ` (Scalar ` w ) ) = ( +g ` R ) )
12		ldualset.a	. . . . . . . . . . 11 |- .+ = ( +g ` R )
13	11, 12	syl6eqr 2674	. . . . . . . . . 10 |- ( w = W -> ( +g ` (Scalar ` w ) ) = .+ )
14		ofeq 6899	. . . . . . . . . 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 5141	. . . . . . . . 9 |- ( w = W -> ( (LFnl ` w ) X. (LFnl ` w ) ) = ( F X. F ) )
17	15, 16	reseq12d 5397	. . . . . . . 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 2674	. . . . . . 7 |- ( w = W -> ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) = .+b )
20	19	opeq2d 4409	. . . . . 6 |- ( w = W -> <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) >. = <. ( +g ` ndx ) , .+b >. )
21	10	fveq2d 6195	. . . . . . . 8 |- ( w = W -> (oppr ` (Scalar ` w ) ) = (oppr ` R ) )
22		ldualset.o	. . . . . . . 8 |- O = (oppr ` R )
23	21, 22	syl6eqr 2674	. . . . . . 7 |- ( w = W -> (oppr ` (Scalar ` w ) ) = O )
24	23	opeq2d 4409	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , (oppr ` (Scalar ` w ) ) >. = <. (Scalar ` ndx ) , O >. )
25	7, 20, 24	tpeq123d 4283	. . . . 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 6195	. . . . . . . . . 10 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` R ) )
27		ldualset.k	. . . . . . . . . 10 |- K = ( Base ` R )
28	26, 27	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
29	10	fveq2d 6195	. . . . . . . . . . . 12 |- ( w = W -> ( .r ` (Scalar ` w ) ) = ( .r ` R ) )
30		ldualset.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
31	29, 30	syl6eqr 2674	. . . . . . . . . . 11 |- ( w = W -> ( .r ` (Scalar ` w ) ) = .x. )
32		ofeq 6899	. . . . . . . . . . 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 2623	. . . . . . . . . 10 |- ( w = W -> f = f )
35		fveq2 6191	. . . . . . . . . . . 12 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
36		ldualset.v	. . . . . . . . . . . 12 |- V = ( Base ` W )
37	35, 36	syl6eqr 2674	. . . . . . . . . . 11 |- ( w = W -> ( Base ` w ) = V )
38	37	xpeq1d 5138	. . . . . . . . . 10 |- ( w = W -> ( ( Base ` w ) X. { k } ) = ( V X. { k } ) )
39	33, 34, 38	oveq123d 6671	. . . . . . . . 9 |- ( w = W -> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) = ( f oF .x. ( V X. { k } ) ) )
40	28, 6, 39	mpt2eq123dv 6717	. . . . . . . 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 2674	. . . . . . 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 4409	. . . . . 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 4189	. . . . 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 3768	. . . 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 34411	. . . 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 6957	. . . . 5 |- { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } e. _V
48		snex 4908	. . . . 5 |- { <. ( .s ` ndx ) , .xb >. } e. _V
49	47, 48	unex 6956	. . . 4 |- ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) e. _V
50	45, 46, 49	fvmpt 6282	. . 3 |- ( W e. _V -> (LDual ` W ) = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )
51	3, 50	syl5eq 2668	. 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 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   {ctp 4181   <.cop 4183   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945  opp_rcoppr 18622  LFnlclfn 34344  LDualcld 34410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ldual 34411

This theorem is referenced by:  ldualvbase  34413  ldualfvadd  34415  ldualsca  34419  ldualfvs  34423