Theorem hdmapval 42580

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 41864). (𝐽‘𝐸) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 42521 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our 𝑧 that the ∀𝑧 ∈ 𝑉 ranges over. The middle term (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩) provides isolation to allow 𝐸 and 𝑇 to assume the same value without conflict. Closure is shown by hdmapcl 42582. If a separate auxiliary vector is known, hdmapval2 42584 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 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))
hdmapval.t	⊢ (𝜑 → 𝑇 ∈ 𝑉)

Assertion

Ref	Expression
hdmapval	⊢ (𝜑 → (𝑆‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 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.

Step	Hyp	Ref	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 → (𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	hdmapfval 42579	. . 3 ⊢ (𝜑 → 𝑆 = (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩)))))
13	12	fveq1d 6885	. 2 ⊢ (𝜑 → (𝑆‘𝑇) = ((𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩))))‘𝑇))
14		hdmapval.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
15		riotaex 7373	. . 3 ⊢ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))) ∈ V
16		sneq 4600	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → {𝑡} = {𝑇})
17	16	fveq2d 6887	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑁‘{𝑡}) = (𝑁‘{𝑇}))
18	17	uneq2d 4123	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) = ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})))
19	18	eleq2d 2849	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) ↔ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))))
20	19	notbid 321	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) ↔ ¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))))
21		oteq3 4850	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩ = ⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)
22	21	fveq2d 6887	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩) = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))
23	22	eqeq2d 2774	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩) ↔ 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩)))
24	20, 23	imbi12d 347	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩)) ↔ (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
25	24	ralbidv 3188	. . . . 5 ⊢ (𝑡 = 𝑇 → (∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩)) ↔ ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
26	25	riotabidv 7371	. . . 4 ⊢ (𝑡 = 𝑇 → (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩))) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
27		eqid 2763	. . . 4 ⊢ (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩)))) = (𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩))))
28	26, 27	fvmptg 6989	. . 3 ⊢ ((𝑇 ∈ 𝑉 ∧ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))) ∈ V) → ((𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩))))‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
29	14, 15, 28	sylancl 597	. 2 ⊢ (𝜑 → ((𝑡 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑡})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑡⟩))))‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))
30	13, 29	eqtrd 2798	1 ⊢ (𝜑 → (𝑆‘𝑇) = (℩𝑦 ∈ 𝐷 ∀𝑧 ∈ 𝑉 (¬ 𝑧 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) → 𝑦 = (𝐼‘⟨𝑧, (𝐼‘⟨𝐸, (𝐽‘𝐸), 𝑧⟩), 𝑇⟩))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∪ cun 3904  {csn 4590  ⟨cop 4596  ⟨cotp 4598   ↦ cmpt 5193   I cid 5557   ↾ cres 5665  ‘cfv 6538  ℩crio 7368  Basecbs 17270  LSpanclspn 21073  LHypclh 40736  LTrncltrn 40853  DVecHcdvh 41830  LCDualclcd 42338  HVMapchvm 42508  HDMap1chdma1 42543  HDMapchdma 42544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-hdmap 42546

This theorem is referenced by:  hdmapcl  42582  hdmapval2lem  42583