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Theorem hdmapfnN 41003
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 = (LHypβ€˜πΎ)
hdmapfn.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hdmapfn.v 𝑉 = (Baseβ€˜π‘ˆ)
hdmapfn.s 𝑆 = ((HDMapβ€˜πΎ)β€˜π‘Š)
hdmapfn.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
Assertion
Ref Expression
hdmapfnN (πœ‘ β†’ 𝑆 Fn 𝑉)

Proof of Theorem hdmapfnN
Dummy variables 𝑦 𝑑 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7371 . . 3 (℩𝑦 ∈ (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š))βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ (((LSpanβ€˜π‘ˆ)β€˜{⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩}) βˆͺ ((LSpanβ€˜π‘ˆ)β€˜{𝑑})) β†’ 𝑦 = (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨π‘§, (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩, (((HVMapβ€˜πΎ)β€˜π‘Š)β€˜βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩), π‘§βŸ©), π‘‘βŸ©))) ∈ V
2 eqid 2730 . . 3 (𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š))βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ (((LSpanβ€˜π‘ˆ)β€˜{⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩}) βˆͺ ((LSpanβ€˜π‘ˆ)β€˜{𝑑})) β†’ 𝑦 = (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨π‘§, (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩, (((HVMapβ€˜πΎ)β€˜π‘Š)β€˜βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩), π‘§βŸ©), π‘‘βŸ©)))) = (𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š))βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ (((LSpanβ€˜π‘ˆ)β€˜{⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩}) βˆͺ ((LSpanβ€˜π‘ˆ)β€˜{𝑑})) β†’ 𝑦 = (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨π‘§, (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩, (((HVMapβ€˜πΎ)β€˜π‘Š)β€˜βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩), π‘§βŸ©), π‘‘βŸ©))))
31, 2fnmpti 6692 . 2 (𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š))βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ (((LSpanβ€˜π‘ˆ)β€˜{⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩}) βˆͺ ((LSpanβ€˜π‘ˆ)β€˜{𝑑})) β†’ 𝑦 = (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨π‘§, (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩, (((HVMapβ€˜πΎ)β€˜π‘Š)β€˜βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩), π‘§βŸ©), π‘‘βŸ©)))) Fn 𝑉
4 hdmapfn.h . . . 4 𝐻 = (LHypβ€˜πΎ)
5 eqid 2730 . . . 4 ⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩ = ⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩
6 hdmapfn.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
7 hdmapfn.v . . . 4 𝑉 = (Baseβ€˜π‘ˆ)
8 eqid 2730 . . . 4 (LSpanβ€˜π‘ˆ) = (LSpanβ€˜π‘ˆ)
9 eqid 2730 . . . 4 ((LCDualβ€˜πΎ)β€˜π‘Š) = ((LCDualβ€˜πΎ)β€˜π‘Š)
10 eqid 2730 . . . 4 (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š)) = (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š))
11 eqid 2730 . . . 4 ((HVMapβ€˜πΎ)β€˜π‘Š) = ((HVMapβ€˜πΎ)β€˜π‘Š)
12 eqid 2730 . . . 4 ((HDMap1β€˜πΎ)β€˜π‘Š) = ((HDMap1β€˜πΎ)β€˜π‘Š)
13 hdmapfn.s . . . 4 𝑆 = ((HDMapβ€˜πΎ)β€˜π‘Š)
14 hdmapfn.k . . . 4 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
154, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hdmapfval 41001 . . 3 (πœ‘ β†’ 𝑆 = (𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š))βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ (((LSpanβ€˜π‘ˆ)β€˜{⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩}) βˆͺ ((LSpanβ€˜π‘ˆ)β€˜{𝑑})) β†’ 𝑦 = (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨π‘§, (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩, (((HVMapβ€˜πΎ)β€˜π‘Š)β€˜βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩), π‘§βŸ©), π‘‘βŸ©)))))
1615fneq1d 6641 . 2 (πœ‘ β†’ (𝑆 Fn 𝑉 ↔ (𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š))βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ (((LSpanβ€˜π‘ˆ)β€˜{⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩}) βˆͺ ((LSpanβ€˜π‘ˆ)β€˜{𝑑})) β†’ 𝑦 = (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨π‘§, (((HDMap1β€˜πΎ)β€˜π‘Š)β€˜βŸ¨βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩, (((HVMapβ€˜πΎ)β€˜π‘Š)β€˜βŸ¨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩), π‘§βŸ©), π‘‘βŸ©)))) Fn 𝑉))
173, 16mpbiri 257 1 (πœ‘ β†’ 𝑆 Fn 𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βˆͺ cun 3945  {csn 4627  βŸ¨cop 4633  βŸ¨cotp 4635   ↦ cmpt 5230   I cid 5572   β†Ύ cres 5677   Fn wfn 6537  β€˜cfv 6542  β„©crio 7366  Basecbs 17148  LSpanclspn 20726  HLchlt 38523  LHypclh 39158  LTrncltrn 39275  DVecHcdvh 40252  LCDualclcd 40760  HVMapchvm 40930  HDMap1chdma1 40965  HDMapchdma 40966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-ot 4636  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-hdmap 40968
This theorem is referenced by:  hdmaprnlem11N  41034  hdmaprnlem17N  41037  hdmaprnN  41038  hdmapf1oN  41039  hgmaprnlem4N  41073
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