Theorem ldualset 35813

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	⊢ ✚ = ( ∘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 3458	. 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		ldualset.d	. . 3 ⊢ 𝐷 = (LDual‘𝑊)
4		fveq2 6545	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = (LFnl‘𝑊))
5		ldualset.f	. . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑊)
6	4, 5	syl6eqr 2851	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LFnl‘𝑤) = 𝐹)
7	6	opeq2d 4723	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), (LFnl‘𝑤)⟩ = ⟨(Base‘ndx), 𝐹⟩)
8		fveq2 6545	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
9		ldualset.r	. . . . . . . . . . . . 13 ⊢ 𝑅 = (Scalar‘𝑊)
10	8, 9	syl6eqr 2851	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝑅)
11	10	fveq2d 6549	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = (+g‘𝑅))
12		ldualset.a	. . . . . . . . . . 11 ⊢ + = (+g‘𝑅)
13	11, 12	syl6eqr 2851	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (+g‘(Scalar‘𝑤)) = + )
14		ofeq 7276	. . . . . . . . . 10 ⊢ ((+g‘(Scalar‘𝑤)) = + → ∘𝑓 (+g‘(Scalar‘𝑤)) = ∘𝑓 + )
15	13, 14	syl 17	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ∘𝑓 (+g‘(Scalar‘𝑤)) = ∘𝑓 + )
16	6	sqxpeqd 5482	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LFnl‘𝑤) × (LFnl‘𝑤)) = (𝐹 × 𝐹))
17	15, 16	reseq12d 5742	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ( ∘𝑓 + ↾ (𝐹 × 𝐹)))
18		ldualset.p	. . . . . . . 8 ⊢ ✚ = ( ∘𝑓 + ↾ (𝐹 × 𝐹))
19	17, 18	syl6eqr 2851	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤))) = ✚ )
20	19	opeq2d 4723	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), ( ∘𝑓 (+g‘(Scalar‘𝑤)) ↾ ((LFnl‘𝑤) × (LFnl‘𝑤)))⟩ = ⟨(+g‘ndx), ✚ ⟩)
21	10	fveq2d 6549	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = (oppr‘𝑅))
22		ldualset.o	. . . . . . . 8 ⊢ 𝑂 = (oppr‘𝑅)
23	21, 22	syl6eqr 2851	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (oppr‘(Scalar‘𝑤)) = 𝑂)
24	23	opeq2d 4723	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), (oppr‘(Scalar‘𝑤))⟩ = ⟨(Scalar‘ndx), 𝑂⟩)
25	7, 20, 24	tpeq123d 4597	. . . . 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⟩})
26	10	fveq2d 6549	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝑅))
27		ldualset.k	. . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝑅)
28	26, 27	syl6eqr 2851	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
29	10	fveq2d 6549	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = (.r‘𝑅))
30		ldualset.t	. . . . . . . . . . . 12 ⊢ · = (.r‘𝑅)
31	29, 30	syl6eqr 2851	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (.r‘(Scalar‘𝑤)) = · )
32		ofeq 7276	. . . . . . . . . . 11 ⊢ ((.r‘(Scalar‘𝑤)) = · → ∘𝑓 (.r‘(Scalar‘𝑤)) = ∘𝑓 · )
33	31, 32	syl 17	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ∘𝑓 (.r‘(Scalar‘𝑤)) = ∘𝑓 · )
34		eqidd 2798	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑓 = 𝑓)
35		fveq2 6545	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
36		ldualset.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑊)
37	35, 36	syl6eqr 2851	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
38	37	xpeq1d 5479	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) × {𝑘}) = (𝑉 × {𝑘}))
39	33, 34, 38	oveq123d 7044	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑓 ∘𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})) = (𝑓 ∘𝑓 · (𝑉 × {𝑘})))
40	28, 6, 39	mpoeq123dv 7094	. . . . . . . 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}))))
41		ldualset.s	. . . . . . . 8 ⊢ ∙ = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘𝑓 · (𝑉 × {𝑘})))
42	40, 41	syl6eqr 2851	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘}))) = ∙ )
43	42	opeq2d 4723	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩ = ⟨( ·𝑠 ‘ndx), ∙ ⟩)
44	43	sneqd 4490	. . . . 5 ⊢ (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑤)), 𝑓 ∈ (LFnl‘𝑤) ↦ (𝑓 ∘𝑓 (.r‘(Scalar‘𝑤))((Base‘𝑤) × {𝑘})))⟩} = {⟨( ·𝑠 ‘ndx), ∙ ⟩})
45	25, 44	uneq12d 4067	. . . 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), ∙ ⟩}))
46		df-ldual 35812	. . . 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})))⟩}))
47		tpex 7334	. . . . 5 ⊢ {⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∈ V
48		snex 5230	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), ∙ ⟩} ∈ V
49	47, 48	unex 7333	. . . 4 ⊢ ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}) ∈ V
50	45, 46, 49	fvmpt 6642	. . 3 ⊢ (𝑊 ∈ V → (LDual‘𝑊) = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
51	3, 50	syl5eq 2845	. 2 ⊢ (𝑊 ∈ V → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))
52	1, 2, 51	3syl 18	1 ⊢ (𝜑 → 𝐷 = ({⟨(Base‘ndx), 𝐹⟩, ⟨(+g‘ndx), ✚ ⟩, ⟨(Scalar‘ndx), 𝑂⟩} ∪ {⟨( ·𝑠 ‘ndx), ∙ ⟩}))

Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  Vcvv 3440   ∪ cun 3863  {csn 4478  {ctp 4482  ⟨cop 4484   × cxp 5448   ↾ cres 5452  ‘cfv 6232  (class class class)co 7023   ∈ cmpo 7025   ∘_𝑓 cof 7272  ndxcnx 16313  Basecbs 16316  +_gcplusg 16398  ._rcmulr 16399  Scalarcsca 16401   ·_𝑠 cvsca 16402  opp_rcoppr 19066  LFnlclfn 35745  LDualcld 35811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-res 5462  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-ldual 35812

This theorem is referenced by:  ldualvbase  35814  ldualfvadd  35816  ldualsca  35820  ldualfvs  35824