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Theorem hdmapval 40502
Description: Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector 𝐸 to be ⟨0, 1⟩ (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom 𝑃 = ((oc‘𝐾)‘𝑊) (see dvheveccl 39786). (𝐽𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 40443 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our 𝑧 that the 𝑧𝑉 ranges over. The middle term (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩) provides isolation to allow 𝐸 and 𝑇 to assume the same value without conflict. Closure is shown by hdmapcl 40504. If a separate auxiliary vector is known, hdmapval2 40506 provides a version without quantification. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmapval.h 𝐻 = (LHyp‘𝐾)
hdmapfval.e 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapfval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapfval.v 𝑉 = (Base‘𝑈)
hdmapfval.n 𝑁 = (LSpan‘𝑈)
hdmapfval.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmapfval.d 𝐷 = (Base‘𝐶)
hdmapfval.j 𝐽 = ((HVMap‘𝐾)‘𝑊)
hdmapfval.i 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmapfval.s 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapfval.k (𝜑 → (𝐾𝐴𝑊𝐻))
hdmapval.t (𝜑𝑇𝑉)
Assertion
Ref Expression
hdmapval (𝜑 → (𝑆𝑇) = (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
Distinct variable groups:   𝑦,𝑧,𝐾   𝑦,𝐷   𝑦,𝐸,𝑧   𝑦,𝐼,𝑧   𝑦,𝑈,𝑧   𝑦,𝑉,𝑧   𝑦,𝑊,𝑧   𝑦,𝑇,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐶(𝑦,𝑧)   𝐷(𝑧)   𝑆(𝑦,𝑧)   𝐻(𝑦,𝑧)   𝐽(𝑦,𝑧)   𝑁(𝑦,𝑧)

Proof of Theorem hdmapval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 hdmapval.h . . . 4 𝐻 = (LHyp‘𝐾)
2 hdmapfval.e . . . 4 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
3 hdmapfval.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
4 hdmapfval.v . . . 4 𝑉 = (Base‘𝑈)
5 hdmapfval.n . . . 4 𝑁 = (LSpan‘𝑈)
6 hdmapfval.c . . . 4 𝐶 = ((LCDual‘𝐾)‘𝑊)
7 hdmapfval.d . . . 4 𝐷 = (Base‘𝐶)
8 hdmapfval.j . . . 4 𝐽 = ((HVMap‘𝐾)‘𝑊)
9 hdmapfval.i . . . 4 𝐼 = ((HDMap1‘𝐾)‘𝑊)
10 hdmapfval.s . . . 4 𝑆 = ((HDMap‘𝐾)‘𝑊)
11 hdmapfval.k . . . 4 (𝜑 → (𝐾𝐴𝑊𝐻))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hdmapfval 40501 . . 3 (𝜑𝑆 = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))))
1312fveq1d 6880 . 2 (𝜑 → (𝑆𝑇) = ((𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))‘𝑇))
14 hdmapval.t . . 3 (𝜑𝑇𝑉)
15 riotaex 7353 . . 3 (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))) ∈ V
16 sneq 4632 . . . . . . . . . . 11 (𝑡 = 𝑇 → {𝑡} = {𝑇})
1716fveq2d 6882 . . . . . . . . . 10 (𝑡 = 𝑇 → (𝑁‘{𝑡}) = (𝑁‘{𝑇}))
1817uneq2d 4159 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) = ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))
1918eleq2d 2818 . . . . . . . 8 (𝑡 = 𝑇 → (𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) ↔ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))))
2019notbid 317 . . . . . . 7 (𝑡 = 𝑇 → (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) ↔ ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))))
21 oteq3 4877 . . . . . . . . 9 (𝑡 = 𝑇 → ⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩)
2221fveq2d 6882 . . . . . . . 8 (𝑡 = 𝑇 → (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))
2322eqeq2d 2742 . . . . . . 7 (𝑡 = 𝑇 → (𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩) ↔ 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩)))
2420, 23imbi12d 344 . . . . . 6 (𝑡 = 𝑇 → ((¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)) ↔ (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
2524ralbidv 3176 . . . . 5 (𝑡 = 𝑇 → (∀𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)) ↔ ∀𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
2625riotabidv 7351 . . . 4 (𝑡 = 𝑇 → (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))) = (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
27 eqid 2731 . . . 4 (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩)))) = (𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))
2826, 27fvmptg 6982 . . 3 ((𝑇𝑉 ∧ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))) ∈ V) → ((𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))‘𝑇) = (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
2914, 15, 28sylancl 586 . 2 (𝜑 → ((𝑡𝑉 ↦ (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑡⟩))))‘𝑇) = (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
3013, 29eqtrd 2771 1 (𝜑 → (𝑆𝑇) = (𝑦𝐷𝑧𝑉𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽𝐸), 𝑧⟩), 𝑇⟩))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3060  Vcvv 3473  cun 3942  {csn 4622  cop 4628  cotp 4630  cmpt 5224   I cid 5566  cres 5671  cfv 6532  crio 7348  Basecbs 17126  LSpanclspn 20531  LHypclh 38658  LTrncltrn 38775  DVecHcdvh 39752  LCDualclcd 40260  HVMapchvm 40430  HDMap1chdma1 40465  HDMapchdma 40466
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-ot 4631  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-hdmap 40468
This theorem is referenced by:  hdmapcl  40504  hdmapval2lem  40505
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