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 = (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 7371	. . 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 2732	. . 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 6693	. 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 2732	. . . 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 2732	. . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
9		eqid 2732	. . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
10		eqid 2732	. . . 4 ⊢ (Base‘((LCDual‘𝐾)‘𝑊)) = (Base‘((LCDual‘𝐾)‘𝑊))
11		eqid 2732	. . . 4 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
12		eqid 2732	. . . 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 41001	. . 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 6642	. 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 257	1 ⊢ (𝜑 → 𝑆 Fn 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∪ cun 3946  {csn 4628  ⟨cop 4634  ⟨cotp 4636   ↦ cmpt 5231   I cid 5573   ↾ cres 5678   Fn wfn 6538  ‘cfv 6543  ℩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 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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-hdmap 40968

This theorem is referenced by:  hdmaprnlem11N  41034  hdmaprnlem17N  41037  hdmaprnN  41038  hdmapf1oN  41039  hgmaprnlem4N  41073