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Theorem hdmapfnN 37438
Description: Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hdmapfn.h 𝐻 = (LHyp‘𝐾)
hdmapfn.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapfn.v 𝑉 = (Base‘𝑈)
hdmapfn.s 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapfn.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
Assertion
Ref Expression
hdmapfnN (𝜑𝑆 Fn 𝑉)

Proof of Theorem hdmapfnN
Dummy variables 𝑦 𝑡 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6655 . . 3 (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑉𝑧 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑡})) → 𝑦 = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑧, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑧⟩), 𝑡⟩))) ∈ V
2 eqid 2651 . . 3 (𝑡𝑉 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑉𝑧 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑡})) → 𝑦 = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑧, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑧⟩), 𝑡⟩)))) = (𝑡𝑉 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑉𝑧 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑡})) → 𝑦 = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑧, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑧⟩), 𝑡⟩))))
31, 2fnmpti 6060 . 2 (𝑡𝑉 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑉𝑧 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑡})) → 𝑦 = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑧, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑧⟩), 𝑡⟩)))) Fn 𝑉
4 hdmapfn.h . . . 4 𝐻 = (LHyp‘𝐾)
5 eqid 2651 . . . 4 ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
6 hdmapfn.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
7 hdmapfn.v . . . 4 𝑉 = (Base‘𝑈)
8 eqid 2651 . . . 4 (LSpan‘𝑈) = (LSpan‘𝑈)
9 eqid 2651 . . . 4 ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
10 eqid 2651 . . . 4 (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
11 eqid 2651 . . . 4 ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
12 eqid 2651 . . . 4 ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊)
13 hdmapfn.s . . . 4 𝑆 = ((HDMap‘𝐾)‘𝑊)
14 hdmapfn.k . . . 4 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
154, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hdmapfval 37436 . . 3 (𝜑𝑆 = (𝑡𝑉 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑉𝑧 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑡})) → 𝑦 = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑧, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑧⟩), 𝑡⟩)))))
1615fneq1d 6019 . 2 (𝜑 → (𝑆 Fn 𝑉 ↔ (𝑡𝑉 ↦ (𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧𝑉𝑧 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑡})) → 𝑦 = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑧, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑧⟩), 𝑡⟩)))) Fn 𝑉))
173, 16mpbiri 248 1 (𝜑𝑆 Fn 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cun 3605  {csn 4210  cop 4216  cotp 4218  cmpt 4762   I cid 5052  cres 5145   Fn wfn 5921  cfv 5926  crio 6650  Basecbs 15904  LSpanclspn 19019  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  DVecHcdvh 36684  LCDualclcd 37192  HVMapchvm 37362  HDMap1chdma1 37398  HDMapchdma 37399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-ot 4219  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-hdmap 37401
This theorem is referenced by:  hdmaprnlem11N  37469  hdmaprnlem17N  37472  hdmaprnN  37473  hdmapf1oN  37474  hgmaprnlem4N  37508
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