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Theorem hdmapfnN 40223
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 7311 . . 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 2737 . . 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 6641 . 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 2737 . . . 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 2737 . . . 4 (LSpan‘𝑈) = (LSpan‘𝑈)
9 eqid 2737 . . . 4 ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
10 eqid 2737 . . . 4 (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
11 eqid 2737 . . . 4 ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
12 eqid 2737 . . . 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 40221 . . 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 6592 . 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 257 1 (𝜑𝑆 Fn 𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3062  cun 3906  {csn 4584  cop 4590  cotp 4592  cmpt 5186   I cid 5528  cres 5633   Fn wfn 6488  cfv 6493  crio 7306  Basecbs 17042  LSpanclspn 20384  HLchlt 37743  LHypclh 38378  LTrncltrn 38495  DVecHcdvh 39472  LCDualclcd 39980  HVMapchvm 40150  HDMap1chdma1 40185  HDMapchdma 40186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-hdmap 40188
This theorem is referenced by:  hdmaprnlem11N  40254  hdmaprnlem17N  40257  hdmaprnN  40258  hdmapf1oN  40259  hgmaprnlem4N  40293
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