Theorem ldualset 39756

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 3478	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		ldualset.d	. . 3 ⊢ 𝐷 = (LDual‘𝑊)
4		fveq2 6871	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5		ldualset.f	. . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑊)
6	4, 5	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
7	6	opeq2d 4840	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8		fveq2 6871	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9		ldualset.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (Scalar‘𝑊)
10	8, 9	eqtr4di 2818	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
11	10	fveq2d 6875	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g‘𝑅))
12		ldualset.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑅)
13	11, 12	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
14	13	ofeqd 7666	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ∘f (+g‘(Scalar‘𝑤)) = ∘f + )
15	6	sqxpeqd 5683	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
16	14, 15	reseq12d 5969	. . . . . . . 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 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ✚ )
19	18	opeq2d 4840	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ✚ ⟩)
20	10	fveq2d 6875	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr‘𝑅))
21		ldualset.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
22	20, 21	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
23	22	opeq2d 4840	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩)
24	7, 19, 23	tpeq123d 4710	. . . . 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 6875	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
26		ldualset.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝑅)
27	25, 26	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
28	10	fveq2d 6875	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r‘𝑅))
29		ldualset.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
30	28, 29	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
31	30	ofeqd 7666	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ∘f (.r‘(Scalar‘𝑤)) = ∘f · )
32		eqidd 2766	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑓 = 𝑓)
33		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
34		ldualset.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑊)
35	33, 34	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
36	35	xpeq1d 5680	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
37	31, 32, 36	oveq123d 7421	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓 ∘f · (𝑉 × {𝑘})))
38	27, 6, 37	mpoeq123dv 7475	. . . . . . . 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 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = ∙ )
41	40	opeq2d 4840	. . . . . 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 4597	. . . . 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 4125	. . . 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 39755	. . . 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 7733	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
46		snex 5400	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), ∙ ⟩} ∈ V
47	45, 46	unex 7731	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}) ∈ V
48	43, 44, 47	fvmpt 6979	. . 3 ⊢ (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
49	3, 48	eqtrid 2812	. 2 ⊢ (𝑊 ∈ V → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
50	1, 2, 49	3syl 19	1 ⊢ (𝜑 → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∪ cun 3905  {csn 4585  {ctp 4589  ⟨cop 4591   × cxp 5649   ↾ cres 5653  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   ∘_f cof 7662  ndxcnx 17241  Basecbs 17257  +_gcplusg 17298  ._rcmulr 17299  Scalarcsca 17301   ·_𝑠 cvsca 17302  opp_rcoppr 20406  LFnlclfn 39688  LDualcld 39754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-ldual 39755

This theorem is referenced by:  ldualvbase  39757  ldualfvadd  39759  ldualsca  39763  ldualfvs  39767