Theorem ldualset 38870

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 to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow 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	⊢ ✚ = ( ∘f + ↾ (𝐹 × 𝐹))
ldualset.f	⊢ 𝐹 = (LFnl‘𝑊)
ldualset.d	⊢ 𝐷 = (LDual‘𝑊)
ldualset.r	⊢ 𝑅 = (Scalar‘𝑊)
ldualset.k	⊢ 𝐾 = (Base‘𝑅)
ldualset.t	⊢ · = (.r‘𝑅)
ldualset.o	⊢ 𝑂 = (oppr‘𝑅)
ldualset.s	⊢ ∙ = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f · (𝑉 × {𝑘})))
ldualset.w	⊢ (𝜑 → 𝑊 ∈ 𝑋)

Assertion

Ref	Expression
ldualset	⊢ (𝜑 → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))

Distinct variable group:   𝑓,𝑘,𝑊

Allowed substitution hints:   𝜑(𝑓,𝑘)   𝐷(𝑓,𝑘)   + (𝑓,𝑘)   ✚ (𝑓,𝑘)   𝑅(𝑓,𝑘)   ∙ (𝑓,𝑘)   · (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 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
2		elex 3493	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		ldualset.d	. . 3 ⊢ 𝐷 = (LDual‘𝑊)
4		fveq2 6905	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5		ldualset.f	. . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑊)
6	4, 5	eqtr4di 2787	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
7	6	opeq2d 4891	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8		fveq2 6905	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9		ldualset.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (Scalar‘𝑊)
10	8, 9	eqtr4di 2787	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
11	10	fveq2d 6909	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g‘𝑅))
12		ldualset.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑅)
13	11, 12	eqtr4di 2787	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
14	13	ofeqd 7697	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ∘f (+g‘(Scalar‘𝑤)) = ∘f + )
15	6	sqxpeqd 5718	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
16	14, 15	reseq12d 5995	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ( ∘f + ↾ (𝐹 × 𝐹)))
17		ldualset.p	. . . . . . . 8 ⊢ ✚ = ( ∘f + ↾ (𝐹 × 𝐹))
18	16, 17	eqtr4di 2787	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ✚ )
19	18	opeq2d 4891	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ✚ ⟩)
20	10	fveq2d 6909	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr‘𝑅))
21		ldualset.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
22	20, 21	eqtr4di 2787	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
23	22	opeq2d 4891	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩)
24	7, 19, 23	tpeq123d 4760	. . . . 5 ⊢ (𝑤 = 𝑊 → {⟨(Base‘ndx), (LFnl‘𝑤)⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩} = {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩})
25	10	fveq2d 6909	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
26		ldualset.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝑅)
27	25, 26	eqtr4di 2787	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
28	10	fveq2d 6909	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r‘𝑅))
29		ldualset.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
30	28, 29	eqtr4di 2787	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
31	30	ofeqd 7697	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ∘f (.r‘(Scalar‘𝑤)) = ∘f · )
32		eqidd 2730	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑓 = 𝑓)
33		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
34		ldualset.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑊)
35	33, 34	eqtr4di 2787	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
36	35	xpeq1d 5715	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
37	31, 32, 36	oveq123d 7451	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓 ∘f · (𝑉 × {𝑘})))
38	27, 6, 37	mpoeq123dv 7506	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f · (𝑉 × {𝑘}))))
39		ldualset.s	. . . . . . . 8 ⊢ ∙ = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f · (𝑉 × {𝑘})))
40	38, 39	eqtr4di 2787	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = ∙ )
41	40	opeq2d 4891	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩ = ⟨( ·𝑠 ‘ndx), ∙ ⟩)
42	41	sneqd 4648	. . . . 5 ⊢ (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩} = {⟨( ·𝑠 ‘ndx), ∙ ⟩})
43	24, 42	uneq12d 4174	. . . 4 ⊢ (𝑤 = 𝑊 → ({⟨(Base‘ndx), (LFnl‘𝑤)⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩}) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
44		df-ldual 38869	. . . 4 ⊢ LDual = (𝑤 ∈ V ↦ ({⟨(Base‘ndx), (LFnl‘𝑤)⟩, ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩, ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩}))
45		tpex 7762	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
46		snex 5441	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), ∙ ⟩} ∈ V
47	45, 46	unex 7760	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}) ∈ V
48	43, 44, 47	fvmpt 7014	. . 3 ⊢ (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
49	3, 48	eqtrid 2781	. 2 ⊢ (𝑊 ∈ V → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
50	1, 2, 49	3syl 18	1 ⊢ (𝜑 → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2100  Vcvv 3472   ∪ cun 3957  {csn 4636  {ctp 4640  ⟨cop 4642   × cxp 5684   ↾ cres 5688  ‘cfv 6558  (class class class)co 7430   ∈ cmpo 7432   ∘_f cof 7693  ndxcnx 17216  Basecbs 17234  +_gcplusg 17287  ._rcmulr 17288  Scalarcsca 17290   ·_𝑠 cvsca 17291  opp_rcoppr 20337  LFnlclfn 38802  LDualcld 38868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5307  ax-nul 5314  ax-pr 5437  ax-un 7751

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3962  df-un 3964  df-in 3966  df-ss 3976  df-nul 4336  df-if 4537  df-sn 4637  df-pr 4639  df-tp 4641  df-op 4643  df-uni 4919  df-br 5157  df-opab 5219  df-mpt 5240  df-id 5584  df-xp 5692  df-rel 5693  df-cnv 5694  df-co 5695  df-dm 5696  df-res 5698  df-iota 6510  df-fun 6560  df-fv 6566  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7695  df-ldual 38869

This theorem is referenced by:  ldualvbase  38871  ldualfvadd  38873  ldualsca  38877  ldualfvs  38881