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Theorem dvhfvadd 36894
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
Hypotheses
Ref Expression
dvhfvadd.h 𝐻 = (LHyp‘𝐾)
dvhfvadd.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhfvadd.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhfvadd.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhfvadd.f 𝐷 = (Scalar‘𝑈)
dvhfvadd.p = (+g𝐷)
dvhfvadd.a = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)
dvhfvadd.s + = (+g𝑈)
Assertion
Ref Expression
dvhfvadd ((𝐾 ∈ HL ∧ 𝑊𝐻) → + = )
Distinct variable groups:   𝑓,𝑔,𝐸   𝑓,𝐻,𝑔   𝑓,𝐾,𝑔   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔
Allowed substitution hints:   𝐷(𝑓,𝑔)   + (𝑓,𝑔)   (𝑓,𝑔)   (𝑓,𝑔)   𝑈(𝑓,𝑔)

Proof of Theorem dvhfvadd
Dummy variables 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhfvadd.h . . . . 5 𝐻 = (LHyp‘𝐾)
2 dvhfvadd.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
3 dvhfvadd.e . . . . 5 𝐸 = ((TEndo‘𝐾)‘𝑊)
4 eqid 2770 . . . . 5 ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
5 dvhfvadd.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
61, 2, 3, 4, 5dvhset 36884 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
76fveq2d 6336 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g𝑈) = (+g‘({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
8 dvhfvadd.p . . . . . . . . . 10 = (+g𝐷)
9 dvhfvadd.f . . . . . . . . . . . 12 𝐷 = (Scalar‘𝑈)
101, 4, 5, 9dvhsca 36885 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
1110fveq2d 6336 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g𝐷) = (+g‘((EDRing‘𝐾)‘𝑊)))
128, 11syl5eq 2816 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → = (+g‘((EDRing‘𝐾)‘𝑊)))
1312oveqd 6809 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((2nd𝑓) (2nd𝑔)) = ((2nd𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd𝑔)))
14133ad2ant1 1126 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd𝑓) (2nd𝑔)) = ((2nd𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd𝑔)))
15 xp2nd 7347 . . . . . . . . . 10 (𝑓 ∈ (𝑇 × 𝐸) → (2nd𝑓) ∈ 𝐸)
16 xp2nd 7347 . . . . . . . . . 10 (𝑔 ∈ (𝑇 × 𝐸) → (2nd𝑔) ∈ 𝐸)
1715, 16anim12i 592 . . . . . . . . 9 ((𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd𝑓) ∈ 𝐸 ∧ (2nd𝑔) ∈ 𝐸))
18 eqid 2770 . . . . . . . . . 10 (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊))
191, 2, 3, 4, 18erngplus 36605 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((2nd𝑓) ∈ 𝐸 ∧ (2nd𝑔) ∈ 𝐸)) → ((2nd𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd𝑔)) = (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘))))
2017, 19sylan2 572 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → ((2nd𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd𝑔)) = (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘))))
21203impb 1106 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd𝑔)) = (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘))))
2214, 21eqtrd 2804 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd𝑓) (2nd𝑔)) = (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘))))
2322opeq2d 4544 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩ = ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)
2423mpt2eq3dva 6865 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩))
25 fvex 6342 . . . . . . . 8 ((LTrn‘𝐾)‘𝑊) ∈ V
262, 25eqeltri 2845 . . . . . . 7 𝑇 ∈ V
27 fvex 6342 . . . . . . . 8 ((TEndo‘𝐾)‘𝑊) ∈ V
283, 27eqeltri 2845 . . . . . . 7 𝐸 ∈ V
2926, 28xpex 7108 . . . . . 6 (𝑇 × 𝐸) ∈ V
3029, 29mpt2ex 7396 . . . . 5 (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩) ∈ V
31 eqid 2770 . . . . . 6 ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}) = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})
3231lmodplusg 16226 . . . . 5 ((𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩) ∈ V → (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩) = (+g‘({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
3330, 32ax-mp 5 . . . 4 (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩) = (+g‘({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
3424, 33syl6req 2821 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩))
357, 34eqtrd 2804 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g𝑈) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩))
36 dvhfvadd.s . 2 + = (+g𝑈)
37 dvhfvadd.a . 2 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)
3835, 36, 373eqtr4g 2829 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → + = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  Vcvv 3349  cun 3719  {csn 4314  {ctp 4318  cop 4320  cmpt 4861   × cxp 5247  ccom 5253  cfv 6031  (class class class)co 6792  cmpt2 6794  1st c1st 7312  2nd c2nd 7313  ndxcnx 16060  Basecbs 16063  +gcplusg 16148  Scalarcsca 16151   ·𝑠 cvsca 16152  HLchlt 35152  LHypclh 35785  LTrncltrn 35902  TEndoctendo 36554  EDRingcedring 36555  DVecHcdvh 36881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-edring 36559  df-dvech 36882
This theorem is referenced by:  dvhvadd  36895
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