Theorem ldualset 39381

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 3461	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		ldualset.d	. . 3 ⊢ 𝐷 = (LDual‘𝑊)
4		fveq2 6834	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5		ldualset.f	. . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑊)
6	4, 5	eqtr4di 2789	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
7	6	opeq2d 4836	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9		ldualset.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (Scalar‘𝑊)
10	8, 9	eqtr4di 2789	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
11	10	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g‘𝑅))
12		ldualset.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑅)
13	11, 12	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
14	13	ofeqd 7624	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ∘f (+g‘(Scalar‘𝑤)) = ∘f + )
15	6	sqxpeqd 5656	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
16	14, 15	reseq12d 5939	. . . . . . . 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 2789	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ✚ )
19	18	opeq2d 4836	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ✚ ⟩)
20	10	fveq2d 6838	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr‘𝑅))
21		ldualset.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
22	20, 21	eqtr4di 2789	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
23	22	opeq2d 4836	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩)
24	7, 19, 23	tpeq123d 4705	. . . . 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 6838	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
26		ldualset.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝑅)
27	25, 26	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
28	10	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r‘𝑅))
29		ldualset.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
30	28, 29	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
31	30	ofeqd 7624	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ∘f (.r‘(Scalar‘𝑤)) = ∘f · )
32		eqidd 2737	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑓 = 𝑓)
33		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
34		ldualset.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑊)
35	33, 34	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
36	35	xpeq1d 5653	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
37	31, 32, 36	oveq123d 7379	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓 ∘f · (𝑉 × {𝑘})))
38	27, 6, 37	mpoeq123dv 7433	. . . . . . . 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 2789	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = ∙ )
41	40	opeq2d 4836	. . . . . 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 4592	. . . . 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 4121	. . . 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 39380	. . . 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 7691	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
46		snex 5381	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), ∙ ⟩} ∈ V
47	45, 46	unex 7689	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}) ∈ V
48	43, 44, 47	fvmpt 6941	. . 3 ⊢ (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
49	3, 48	eqtrid 2783	. 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 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899  {csn 4580  {ctp 4584  ⟨cop 4586   × cxp 5622   ↾ cres 5626  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   ∘_f cof 7620  ndxcnx 17120  Basecbs 17136  +_gcplusg 17177  ._rcmulr 17178  Scalarcsca 17180   ·_𝑠 cvsca 17181  opp_rcoppr 20272  LFnlclfn 39313  LDualcld 39379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-ldual 39380

This theorem is referenced by:  ldualvbase  39382  ldualfvadd  39384  ldualsca  39388  ldualfvs  39392