Theorem dvaset 41504

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 41503	. . . 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 6836	. . 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 6834	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
6		dvaset.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7	5, 6	eqtr4di 2793	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
8	7	opeq2d 4818	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), 𝑇⟩)
9		eqidd 2741	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
10	7, 7, 9	mpoeq123dv 7438	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)))
11	10	opeq2d 4818	. . . . . 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 6834	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = ((EDRing‘𝐾)‘𝑊))
13		dvaset.d	. . . . . . . 8 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)
14	12, 13	eqtr4di 2793	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((EDRing‘𝐾)‘𝑤) = 𝐷)
15	14	opeq2d 4818	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩ = ⟨(Scalar‘ndx), 𝐷⟩)
16	8, 11, 15	tpeq123d 4687	. . . . 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 6834	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
18		dvaset.e	. . . . . . . . 9 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
19	17, 18	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
20		eqidd 2741	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑠‘𝑓) = (𝑠‘𝑓))
21	19, 7, 20	mpoeq123dv 7438	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠‘𝑓)) = (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓)))
22	21	opeq2d 4818	. . . . . 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 4574	. . . . 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 4106	. . . 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 2740	. . . 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 7696	. . . . 5 ⊢ {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∈ V
27		snex 5375	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))⟩} ∈ V
28	26, 27	unex 7694	. . . 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 6942	. . 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 2795	. 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 2787	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 396   = wceq 1547   ∈ wcel 2119   ∪ cun 3888  {csn 4562  {ctp 4566  ⟨cop 4568   ↦ cmpt 5160   ∘ ccom 5629  ‘cfv 6492   ∈ cmpo 7365  ndxcnx 17161  Basecbs 17177  +_gcplusg 17218  Scalarcsca 17221   ·_𝑠 cvsca 17222  LHypclh 40483  LTrncltrn 40600  TEndoctendo 41251  EDRingcedring 41252  DVecAcdveca 41501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-oprab 7367  df-mpo 7368  df-dveca 41502

This theorem is referenced by:  dvasca  41505  dvavbase  41512  dvafvadd  41513  dvafvsca  41515  dvaabl  41523