Theorem dvaset 41124

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

Hypotheses

Ref	Expression
dvaset.h	⊢ 𝐻 = (LHyp‘𝐾)
dvaset.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvaset.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvaset.d	⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)
dvaset.u	⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)

Assertion

Ref	Expression
dvaset	⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩}))

Distinct variable groups:   𝑓,𝑔,𝑠,𝐾   𝑓,𝑊,𝑔,𝑠

Allowed substitution hints:   𝐷(𝑓,𝑔,𝑠)   𝑇(𝑓,𝑔,𝑠)   𝑈(𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔,𝑠)   𝐻(𝑓,𝑔,𝑠)   𝑋(𝑓,𝑔,𝑠)

Proof of Theorem dvaset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvaset.u	. 2 ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)
2		dvaset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3	2	dvafset 41123	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (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))⟩})))
4	3	fveq1d 6830	. . 3 ⊢ (𝐾 ∈ 𝑋 → ((DVecA‘𝐾)‘𝑊) = ((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))⟩}))‘𝑊))
5		fveq2 6828	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
6		dvaset.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7	5, 6	eqtr4di 2786	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
8	7	opeq2d 4831	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), 𝑇⟩)
9		eqidd 2734	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
10	7, 7, 9	mpoeq123dv 7427	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)))
11	10	opeq2d 4831	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩)
12		fveq2 6828	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
13		dvaset.d	. . . . . . . 8 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)
14	12, 13	eqtr4di 2786	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = 𝐷)
15	14	opeq2d 4831	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩ = ⟨(Scalar‘ndx), 𝐷⟩)
16	8, 11, 15	tpeq123d 4700	. . . . 5 ⊢ (𝑤 = 𝑊 → {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩})
17		fveq2 6828	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
18		dvaset.e	. . . . . . . . 9 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
19	17, 18	eqtr4di 2786	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
20		eqidd 2734	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑠‘𝑓) = (𝑠‘𝑓))
21	19, 7, 20	mpoeq123dv 7427	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓)) = (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓)))
22	21	opeq2d 4831	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩ = ⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩)
23	22	sneqd 4587	. . . . 5 ⊢ (𝑤 = 𝑊 → {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓))⟩} = {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩})
24	16, 23	uneq12d 4118	. . . 4 ⊢ (𝑤 = 𝑊 → ({⟨(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), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩}))
25		eqid 2733	. . . 4 ⊢ (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))⟩})) = (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))⟩}))
26		tpex 7685	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∈ V
27		snex 5376	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩} ∈ V
28	26, 27	unex 7683	. . . 4 ⊢ ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩}) ∈ V
29	24, 25, 28	fvmpt 6935	. . 3 ⊢ (𝑊 ∈ 𝐻 → ((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))⟩}))‘𝑊) = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩}))
30	4, 29	sylan9eq 2788	. 2 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → ((DVecA‘𝐾)‘𝑊) = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩}))
31	1, 30	eqtrid 2780	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∪ cun 3896  {csn 4575  {ctp 4579  ⟨cop 4581   ↦ cmpt 5174   ∘ ccom 5623  ‘cfv 6486   ∈ cmpo 7354  ndxcnx 17106  Basecbs 17122  +_gcplusg 17163  Scalarcsca 17166   ·_𝑠 cvsca 17167  LHypclh 40103  LTrncltrn 40220  TEndoctendo 40871  EDRingcedring 40872  DVecAcdveca 41121

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-oprab 7356  df-mpo 7357  df-dveca 41122

This theorem is referenced by:  dvasca  41125  dvavbase  41132  dvafvadd  41133  dvafvsca  41135  dvaabl  41143