Theorem dvafset 38300

Description: The constructed partial vector space A for a lattice 𝐾. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)

Hypothesis

Ref	Expression
dvaset.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
dvafset	⊢ (𝐾 ∈ 𝑉 → (DVecA‘𝐾) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩})))

Distinct variable groups:   𝑤,𝐻   𝑓,𝑔,𝑠,𝑤,𝐾

Allowed substitution hints:   𝐻(𝑓,𝑔,𝑠)   𝑉(𝑤,𝑓,𝑔,𝑠)

Proof of Theorem dvafset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3459	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6645	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		dvaset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2851	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6645	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
6	5	fveq1d 6647	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
7	6	opeq2d 4772	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩ = ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩)
8		eqidd 2799	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
9	6, 6, 8	mpoeq123dv 7208	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔)))
10	9	opeq2d 4772	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩)
11		fveq2 6645	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (EDRing‘𝑘) = (EDRing‘𝐾))
12	11	fveq1d 6647	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((EDRing‘𝑘)‘𝑤) = ((EDRing‘𝐾)‘𝑤))
13	12	opeq2d 4772	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩ = ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩)
14	7, 10, 13	tpeq123d 4644	. . . . 5 ⊢ (𝑘 = 𝐾 → {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} = {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩})
15		fveq2 6645	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
16	15	fveq1d 6647	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
17		eqidd 2799	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑠‘𝑓) = (𝑠‘𝑓))
18	16, 6, 17	mpoeq123dv 7208	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠‘𝑓)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓)))
19	18	opeq2d 4772	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠‘𝑓))⟩ = ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩)
20	19	sneqd 4537	. . . . 5 ⊢ (𝑘 = 𝐾 → {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠‘𝑓))⟩} = {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩})
21	14, 20	uneq12d 4091	. . . 4 ⊢ (𝑘 = 𝐾 → ({⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠‘𝑓))⟩}) = ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩}))
22	4, 21	mpteq12dv 5115	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠‘𝑓))⟩})) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩})))
23		df-dveca 38299	. . 3 ⊢ DVecA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠‘𝑓))⟩})))
24	22, 23, 3	mptfvmpt 6968	. 2 ⊢ (𝐾 ∈ V → (DVecA‘𝐾) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩})))
25	1, 24	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → (DVecA‘𝐾) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩})))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∪ cun 3879  {csn 4525  {ctp 4529  ⟨cop 4531   ↦ cmpt 5110   ∘ ccom 5523  ‘cfv 6324   ∈ cmpo 7137  ndxcnx 16472  Basecbs 16475  +_gcplusg 16557  Scalarcsca 16560   ·_𝑠 cvsca 16561  LHypclh 37280  LTrncltrn 37397  TEndoctendo 38048  EDRingcedring 38049  DVecAcdveca 38298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-oprab 7139  df-mpo 7140  df-dveca 38299

This theorem is referenced by:  dvaset  38301