Theorem hdmapfnN 41831

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 ∧ 𝑊 ∈ 𝐻))

Assertion

Ref	Expression
hdmapfnN	⊢ (𝜑 → 𝑆 Fn 𝑉)

Proof of Theorem hdmapfnN

Dummy variables 𝑦 𝑡 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 7392	. . 3 ⊢ (℩𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (((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 ⊢ (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (((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⟩)))) = (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (((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⟩))))
3	1, 2	fnmpti 6711	. 2 ⊢ (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (((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 ∧ 𝑊 ∈ 𝐻))
15	4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	hdmapfval 41829	. . 3 ⊢ (𝜑 → 𝑆 = (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (((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⟩)))))
16	15	fneq1d 6661	. 2 ⊢ (𝜑 → (𝑆 Fn 𝑉 ↔ (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝐾)‘𝑊))∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (((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))
17	3, 16	mpbiri 258	1 ⊢ (𝜑 → 𝑆 Fn 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ∪ cun 3949  {csn 4626  ⟨cop 4632  ⟨cotp 4634   ↦ cmpt 5225   I cid 5577   ↾ cres 5687   Fn wfn 6556  ‘cfv 6561  ℩crio 7387  Basecbs 17247  LSpanclspn 20969  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  DVecHcdvh 41080  LCDualclcd 41588  HVMapchvm 41758  HDMap1chdma1 41793  HDMapchdma 41794

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-hdmap 41796

This theorem is referenced by:  hdmaprnlem11N  41862  hdmaprnlem17N  41865  hdmaprnN  41866  hdmapf1oN  41867  hgmaprnlem4N  41901