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Theorem hdmapfnN 36139
 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 6515 . . 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 2610 . . 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 5935 . 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 2610 . . . 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 2610 . . . 4 (LSpan‘𝑈) = (LSpan‘𝑈)
9 eqid 2610 . . . 4 ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
10 eqid 2610 . . . 4 (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
11 eqid 2610 . . . 4 ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
12 eqid 2610 . . . 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 36137 . . 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 5895 . 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 247 1 (𝜑𝑆 Fn 𝑉)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538  {csn 4125  ⟨cop 4131  ⟨cotp 4133   ↦ cmpt 4643   I cid 4948   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804  ℩crio 6510  Basecbs 15695  LSpanclspn 18792  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  DVecHcdvh 35385  LCDualclcd 35893  HVMapchvm 36063  HDMap1chdma1 36099  HDMapchdma 36100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-hdmap 36102 This theorem is referenced by:  hdmaprnlem11N  36170  hdmaprnlem17N  36173  hdmaprnN  36174  hdmapf1oN  36175  hgmaprnlem4N  36209
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