Theorem ldualset 32775

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 2986	. 2 |- ( W e. X -> W e. _V )
3		ldualset.d	. . 3 |- D = (LDual ` W )
4		fveq2 5696	. . . . . . . 8 |- ( w = W -> (LFnl ` w ) = (LFnl ` W ) )
5		ldualset.f	. . . . . . . 8 |- F = (LFnl ` W )
6	4, 5	syl6eqr 2493	. . . . . . 7 |- ( w = W -> (LFnl ` w ) = F )
7	6	opeq2d 4071	. . . . . 6 |- ( w = W -> <. ( Base ` ndx ) , (LFnl ` w ) >. = <. ( Base ` ndx ) , F >. )
8		fveq2 5696	. . . . . . . . . . . . 13 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
9		ldualset.r	. . . . . . . . . . . . 13 |- R = (Scalar ` W )
10	8, 9	syl6eqr 2493	. . . . . . . . . . . 12 |- ( w = W -> (Scalar ` w ) = R )
11	10	fveq2d 5700	. . . . . . . . . . 11 |- ( w = W -> ( +g ` (Scalar ` w ) ) = ( +g ` R ) )
12		ldualset.a	. . . . . . . . . . 11 |- .+ = ( +g ` R )
13	11, 12	syl6eqr 2493	. . . . . . . . . 10 |- ( w = W -> ( +g ` (Scalar ` w ) ) = .+ )
14		ofeq 6327	. . . . . . . . . 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 4870	. . . . . . . . 9 |- ( w = W -> ( (LFnl ` w ) X. (LFnl ` w ) ) = ( F X. F ) )
17	15, 16	reseq12d 5116	. . . . . . . 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 2493	. . . . . . 7 |- ( w = W -> ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) = .+b )
20	19	opeq2d 4071	. . . . . 6 |- ( w = W -> <. ( +g ` ndx ) , ( oF ( +g ` (Scalar ` w ) ) |` ( (LFnl ` w ) X. (LFnl ` w ) ) ) >. = <. ( +g ` ndx ) , .+b >. )
21	10	fveq2d 5700	. . . . . . . 8 |- ( w = W -> (oppr ` (Scalar ` w ) ) = (oppr ` R ) )
22		ldualset.o	. . . . . . . 8 |- O = (oppr ` R )
23	21, 22	syl6eqr 2493	. . . . . . 7 |- ( w = W -> (oppr ` (Scalar ` w ) ) = O )
24	23	opeq2d 4071	. . . . . 6 |- ( w = W -> <. (Scalar ` ndx ) , (oppr ` (Scalar ` w ) ) >. = <. (Scalar ` ndx ) , O >. )
25	7, 20, 24	tpeq123d 3974	. . . . 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 5700	. . . . . . . . . 10 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` R ) )
27		ldualset.k	. . . . . . . . . 10 |- K = ( Base ` R )
28	26, 27	syl6eqr 2493	. . . . . . . . 9 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
29	10	fveq2d 5700	. . . . . . . . . . . 12 |- ( w = W -> ( .r ` (Scalar ` w ) ) = ( .r ` R ) )
30		ldualset.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
31	29, 30	syl6eqr 2493	. . . . . . . . . . 11 |- ( w = W -> ( .r ` (Scalar ` w ) ) = .x. )
32		ofeq 6327	. . . . . . . . . . 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 2444	. . . . . . . . . 10 |- ( w = W -> f = f )
35		fveq2 5696	. . . . . . . . . . . 12 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
36		ldualset.v	. . . . . . . . . . . 12 |- V = ( Base ` W )
37	35, 36	syl6eqr 2493	. . . . . . . . . . 11 |- ( w = W -> ( Base ` w ) = V )
38	37	xpeq1d 4868	. . . . . . . . . 10 |- ( w = W -> ( ( Base ` w ) X. { k } ) = ( V X. { k } ) )
39	33, 34, 38	oveq123d 6117	. . . . . . . . 9 |- ( w = W -> ( f oF ( .r ` (Scalar ` w ) ) ( ( Base ` w ) X. { k } ) ) = ( f oF .x. ( V X. { k } ) ) )
40	28, 6, 39	mpt2eq123dv 6153	. . . . . . . 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 2493	. . . . . . 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 4071	. . . . . 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 3894	. . . . 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 3516	. . . 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 32774	. . . 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 6384	. . . . 5 |- { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } e. _V
48		snex 4538	. . . . 5 |- { <. ( .s ` ndx ) , .xb >. } e. _V
49	47, 48	unex 6383	. . . 4 |- ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) e. _V
50	45, 46, 49	fvmpt 5779	. . 3 |- ( W e. _V -> (LDual ` W ) = ( { <. ( Base ` ndx ) , F >. , <. ( +g ` ndx ) , .+b >. , <. (Scalar ` ndx ) , O >. } u. { <. ( .s ` ndx ) , .xb >. } ) )
51	3, 50	syl5eq 2487	. 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 1369   e. wcel 1756   _Vcvv 2977   u. cun 3331   {csn 3882   {ctp 3886   <.cop 3888   X. cxp 4843   |` cres 4847   ` cfv 5423  (class class class)co 6096   e. cmpt2 6098   oFcof 6323   ndxcnx 14176   Basecbs 14179   +g cplusg 14243   .rcmulr 14244  Scalarcsca 14246   .scvsca 14247  opp_rcoppr 16719  LFnlclfn 32707  LDualcld 32773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-ldual 32774

This theorem is referenced by:  ldualvbase  32776  ldualfvadd  32778  ldualsca  32782  ldualfvs  32786