Theorem mapdffval 42507

Description: Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)

Hypothesis

Ref	Expression
mapdval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
mapdffval	⊢ (𝐾 ∈ 𝑋 → (mapd‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})))

Distinct variable groups:   𝑤,𝐻   𝑓,𝑠,𝑤,𝐾

Allowed substitution hints:   𝐻(𝑓, 𝑠)   𝑋(𝑤, 𝑓, 𝑠)

Proof of Theorem mapdffval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3474	. 2 ⊢ (𝐾 ∈ 𝑋 → 𝐾 ∈ V)
2		fveq2 6882	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		mapdval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2815	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6882	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
6	5	fveq1d 6884	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
7	6	fveq2d 6886	. . . . 5 ⊢ (𝑘 = 𝐾 → (LSubSp‘((DVecH‘𝑘)‘𝑤)) = (LSubSp‘((DVecH‘𝐾)‘𝑤)))
8	6	fveq2d 6886	. . . . . 6 ⊢ (𝑘 = 𝐾 → (LFnl‘((DVecH‘𝑘)‘𝑤)) = (LFnl‘((DVecH‘𝐾)‘𝑤)))
9		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (ocH‘𝑘) = (ocH‘𝐾))
10	9	fveq1d 6884	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((ocH‘𝑘)‘𝑤) = ((ocH‘𝐾)‘𝑤))
11	6	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (LKer‘((DVecH‘𝑘)‘𝑤)) = (LKer‘((DVecH‘𝐾)‘𝑤)))
12	11	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))
13	10, 12	fveq12d 6889	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) = (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)))
14	10, 13	fveq12d 6889	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))))
15	14, 12	eqeq12d 2778	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ↔ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)))
16	13	sseq1d 3965	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠 ↔ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠))
17	15, 16	anbi12d 644	. . . . . 6 ⊢ (𝑘 = 𝐾 → (((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠) ↔ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)))
18	8, 17	rabeqbidv 3432	. . . . 5 ⊢ (𝑘 = 𝐾 → {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠)} = {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})
19	7, 18	mpteq12dv 5196	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑠 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠)}) = (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)}))
20	4, 19	mpteq12dv 5196	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠)})) = (𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})))
21		df-mapd 42506	. . 3 ⊢ mapd = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠)})))
22	20, 21, 3	mptfvmpt 7231	. 2 ⊢ (𝐾 ∈ V → (mapd‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})))
23	1, 22	syl 18	1 ⊢ (𝐾 ∈ 𝑋 → (mapd‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ⊆ wss 3902   ↦ cmpt 5190  ‘cfv 6537  LSubSpclss 21121  LFnlclfn 39938  LKerclk 39966  LHypclh 40865  DVecHcdvh 41959  ocHcoch 42228  mapdcmpd 42505

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-mapd 42506

This theorem is used by:  mapdfval  42508