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Theorem mapdfval 39378
Description: Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
Hypotheses
Ref Expression
mapdval.h 𝐻 = (LHyp‘𝐾)
mapdval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdval.s 𝑆 = (LSubSp‘𝑈)
mapdval.f 𝐹 = (LFnl‘𝑈)
mapdval.l 𝐿 = (LKer‘𝑈)
mapdval.o 𝑂 = ((ocH‘𝐾)‘𝑊)
mapdval.m 𝑀 = ((mapd‘𝐾)‘𝑊)
Assertion
Ref Expression
mapdfval ((𝐾𝑋𝑊𝐻) → 𝑀 = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
Distinct variable groups:   𝑓,𝑠,𝐾   𝑓,𝐹   𝑆,𝑠   𝑓,𝑊,𝑠
Allowed substitution hints:   𝑆(𝑓)   𝑈(𝑓,𝑠)   𝐹(𝑠)   𝐻(𝑓,𝑠)   𝐿(𝑓,𝑠)   𝑀(𝑓,𝑠)   𝑂(𝑓,𝑠)   𝑋(𝑓,𝑠)

Proof of Theorem mapdfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 mapdval.m . . 3 𝑀 = ((mapd‘𝐾)‘𝑊)
2 mapdval.h . . . . 5 𝐻 = (LHyp‘𝐾)
32mapdffval 39377 . . . 4 (𝐾𝑋 → (mapd‘𝐾) = (𝑤𝐻 ↦ (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)})))
43fveq1d 6719 . . 3 (𝐾𝑋 → ((mapd‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (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))
51, 4syl5eq 2790 . 2 (𝐾𝑋𝑀 = ((𝑤𝐻 ↦ (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))
6 fveq2 6717 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
7 mapdval.u . . . . . . 7 𝑈 = ((DVecH‘𝐾)‘𝑊)
86, 7eqtr4di 2796 . . . . . 6 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
98fveq2d 6721 . . . . 5 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈))
10 mapdval.s . . . . 5 𝑆 = (LSubSp‘𝑈)
119, 10eqtr4di 2796 . . . 4 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
128fveq2d 6721 . . . . . 6 (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = (LFnl‘𝑈))
13 mapdval.f . . . . . 6 𝐹 = (LFnl‘𝑈)
1412, 13eqtr4di 2796 . . . . 5 (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = 𝐹)
15 fveq2 6717 . . . . . . . . 9 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
16 mapdval.o . . . . . . . . 9 𝑂 = ((ocH‘𝐾)‘𝑊)
1715, 16eqtr4di 2796 . . . . . . . 8 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = 𝑂)
188fveq2d 6721 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = (LKer‘𝑈))
19 mapdval.l . . . . . . . . . . 11 𝐿 = (LKer‘𝑈)
2018, 19eqtr4di 2796 . . . . . . . . . 10 (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = 𝐿)
2120fveq1d 6719 . . . . . . . . 9 (𝑤 = 𝑊 → ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) = (𝐿𝑓))
2217, 21fveq12d 6724 . . . . . . . 8 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) = (𝑂‘(𝐿𝑓)))
2317, 22fveq12d 6724 . . . . . . 7 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = (𝑂‘(𝑂‘(𝐿𝑓))))
2423, 21eqeq12d 2753 . . . . . 6 (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ↔ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)))
2522sseq1d 3932 . . . . . 6 (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿𝑓)) ⊆ 𝑠))
2624, 25anbi12d 634 . . . . 5 (𝑤 = 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) ↔ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)))
2714, 26rabeqbidv 3396 . . . 4 (𝑤 = 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)} = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})
2811, 27mpteq12dv 5140 . . 3 (𝑤 = 𝑊 → (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)}) = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
29 eqid 2737 . . 3 (𝑤𝐻 ↦ (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)})) = (𝑤𝐻 ↦ (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)}))
3028, 29, 10mptfvmpt 7044 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (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) = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
315, 30sylan9eq 2798 1 ((𝐾𝑋𝑊𝐻) → 𝑀 = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  {crab 3065  wss 3866  cmpt 5135  cfv 6380  LSubSpclss 19968  LFnlclfn 36808  LKerclk 36836  LHypclh 37735  DVecHcdvh 38829  ocHcoch 39098  mapdcmpd 39375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-mapd 39376
This theorem is referenced by:  mapdval  39379  mapd1o  39399
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