Theorem dvhsca 41884

Description: The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)

Hypotheses

Ref	Expression
dvhsca.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhsca.d	⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)
dvhsca.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhsca.f	⊢ 𝐹 = (Scalar‘𝑈)

Assertion

Ref	Expression
dvhsca	⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐹 = 𝐷)

Proof of Theorem dvhsca

Dummy variables 𝑓 𝑔 ℎ 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhsca.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2762	. . . 4 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
3		eqid 2762	. . . 4 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
4		dvhsca.d	. . . 4 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)
5		dvhsca.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dvhset 41883	. . 3 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩}))
7	6	fveq2d 6885	. 2 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → (Scalar‘𝑈) = (Scalar‘({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})))
8		dvhsca.f	. 2 ⊢ 𝐹 = (Scalar‘𝑈)
9	4	fvexi 6895	. . 3 ⊢ 𝐷 ∈ V
10		eqid 2762	. . . 4 ⊢ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩}) = ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})
11	10	lmodsca 17387	. . 3 ⊢ (𝐷 ∈ V → 𝐷 = (Scalar‘({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})))
12	9, 11	ax-mp 5	. 2 ⊢ 𝐷 = (Scalar‘({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ (((LTrn‘𝐾)‘𝑊) × ((TEndo‘𝐾)‘𝑊)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩}))
13	7, 8, 12	3eqtr4g 2822	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐹 = 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ∪ cun 3902  {csn 4588  {ctp 4592  ⟨cop 4594   ↦ cmpt 5191   × cxp 5658   ∘ ccom 5664  ‘cfv 6536   ∈ cmpo 7414  1^st c1st 7982  2^nd c2nd 7983  ndxcnx 17259  Basecbs 17275  +_gcplusg 17316  Scalarcsca 17319   ·_𝑠 cvsca 17320  LHypclh 40786  LTrncltrn 40903  TEndoctendo 41554  EDRingcedring 41555  DVecHcdvh 41880

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-n0 12511  df-z 12598  df-uz 12869  df-fz 13542  df-struct 17213  df-slot 17248  df-ndx 17260  df-base 17276  df-plusg 17329  df-sca 17332  df-vsca 17333  df-dvech 41881

This theorem is used by:  dvhbase  41885  dvhfplusr  41886  dvhfmulr  41887  dvhfvadd  41893  dvhvaddass  41899  tendoinvcl  41906  tendolinv  41907  tendorinv  41908  dvhgrp  41909  dvhlveclem  41910  cdlemn4  42000  hlhilsbase2  42744  hlhilsplus2  42745  hlhilsmul2  42746