Theorem hdmapfnN 42767

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 7377	. . 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 2760	. . 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 6678	. 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 2760	. . . 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 2760	. . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
9		eqid 2760	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
10		eqid 2760	. . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
11		eqid 2760	. . . 4 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
12		eqid 2760	. . . 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 42765	. . 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 6628	. 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 261	1 ⊢ (𝜑 → 𝑆 Fn 𝑉)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∪ cun 3897  {csn 4584  ⟨cop 4590  ⟨cotp 4592   ↦ cmpt 5186   I cid 5549   ↾ cres 5657   Fn wfn 6530  ‘cfv 6535  ℩crio 7372  Basecbs 17326  LSpanclspn 21185  HLchlt 40288  LHypclh 40922  LTrncltrn 41039  DVecHcdvh 42016  LCDualclcd 42524  HVMapchvm 42694  HDMap1chdma1 42729  HDMapchdma 42730

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-hdmap 42732

This theorem is used by:  hdmaprnlem11N  42798  hdmaprnlem17N  42801  hdmaprnN  42802  hdmapf1oN  42803  hgmaprnlem4N  42837