Theorem ldualset 39713

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 3474	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		ldualset.d	. . 3 ⊢ 𝐷 = (LDual‘𝑊)
4		fveq2 6863	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5		ldualset.f	. . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑊)
6	4, 5	eqtr4di 2814	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
7	6	opeq2d 4837	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8		fveq2 6863	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9		ldualset.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (Scalar‘𝑊)
10	8, 9	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
11	10	fveq2d 6867	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g‘𝑅))
12		ldualset.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑅)
13	11, 12	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
14	13	ofeqd 7658	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ∘f (+g‘(Scalar‘𝑤)) = ∘f + )
15	6	sqxpeqd 5677	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
16	14, 15	reseq12d 5964	. . . . . . . 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 2814	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ✚ )
19	18	opeq2d 4837	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘f (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ✚ ⟩)
20	10	fveq2d 6867	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr‘𝑅))
21		ldualset.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
22	20, 21	eqtr4di 2814	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
23	22	opeq2d 4837	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩)
24	7, 19, 23	tpeq123d 4706	. . . . 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 6867	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
26		ldualset.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝑅)
27	25, 26	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
28	10	fveq2d 6867	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r‘𝑅))
29		ldualset.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
30	28, 29	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
31	30	ofeqd 7658	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ∘f (.r‘(Scalar‘𝑤)) = ∘f · )
32		eqidd 2762	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑓 = 𝑓)
33		fveq2 6863	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
34		ldualset.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑊)
35	33, 34	eqtr4di 2814	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
36	35	xpeq1d 5674	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
37	31, 32, 36	oveq123d 7413	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓 ∘f · (𝑉 × {𝑘})))
38	27, 6, 37	mpoeq123dv 7467	. . . . . . . 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 2814	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘f (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = ∙ )
41	40	opeq2d 4837	. . . . . 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 4593	. . . . 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 4122	. . . 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 39712	. . . 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 7725	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
46		snex 5395	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), ∙ ⟩} ∈ V
47	45, 46	unex 7723	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}) ∈ V
48	43, 44, 47	fvmpt 6971	. . 3 ⊢ (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
49	3, 48	eqtrid 2808	. 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 1559   ∈ wcel 2141  Vcvv 3453   ∪ cun 3902  {csn 4581  {ctp 4585  ⟨cop 4587   × cxp 5643   ↾ cres 5647  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394   ∘_f cof 7654  ndxcnx 17212  Basecbs 17228  +_gcplusg 17269  ._rcmulr 17270  Scalarcsca 17272   ·_𝑠 cvsca 17273  opp_rcoppr 20364  LFnlclfn 39645  LDualcld 39711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-res 5657  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-ldual 39712

This theorem is referenced by:  ldualvbase  39714  ldualfvadd  39716  ldualsca  39720  ldualfvs  39724