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Theorem mapdval 41630
Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 27-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)
mapdval.k (𝜑 → (𝐾𝑋𝑊𝐻))
mapdval.t (𝜑𝑇𝑆)
Assertion
Ref Expression
mapdval (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
Distinct variable groups:   𝑓,𝐾   𝑓,𝐹   𝑓,𝑊   𝑇,𝑓
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝑈(𝑓)   𝐻(𝑓)   𝐿(𝑓)   𝑀(𝑓)   𝑂(𝑓)   𝑋(𝑓)

Proof of Theorem mapdval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 mapdval.k . . . 4 (𝜑 → (𝐾𝑋𝑊𝐻))
2 mapdval.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 mapdval.u . . . . 5 𝑈 = ((DVecH‘𝐾)‘𝑊)
4 mapdval.s . . . . 5 𝑆 = (LSubSp‘𝑈)
5 mapdval.f . . . . 5 𝐹 = (LFnl‘𝑈)
6 mapdval.l . . . . 5 𝐿 = (LKer‘𝑈)
7 mapdval.o . . . . 5 𝑂 = ((ocH‘𝐾)‘𝑊)
8 mapdval.m . . . . 5 𝑀 = ((mapd‘𝐾)‘𝑊)
92, 3, 4, 5, 6, 7, 8mapdfval 41629 . . . 4 ((𝐾𝑋𝑊𝐻) → 𝑀 = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
101, 9syl 17 . . 3 (𝜑𝑀 = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}))
1110fveq1d 6908 . 2 (𝜑 → (𝑀𝑇) = ((𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})‘𝑇))
12 mapdval.t . . 3 (𝜑𝑇𝑆)
135fvexi 6920 . . . 4 𝐹 ∈ V
1413rabex 5339 . . 3 {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)} ∈ V
15 sseq2 4010 . . . . . 6 (𝑠 = 𝑇 → ((𝑂‘(𝐿𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿𝑓)) ⊆ 𝑇))
1615anbi2d 630 . . . . 5 (𝑠 = 𝑇 → (((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠) ↔ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)))
1716rabbidv 3444 . . . 4 (𝑠 = 𝑇 → {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)} = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
18 eqid 2737 . . . 4 (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)}) = (𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})
1917, 18fvmptg 7014 . . 3 ((𝑇𝑆 ∧ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)} ∈ V) → ((𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})‘𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
2012, 14, 19sylancl 586 . 2 (𝜑 → ((𝑠𝑆 ↦ {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑠)})‘𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
2111, 20eqtrd 2777 1 (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  cmpt 5225  cfv 6561  LSubSpclss 20929  LFnlclfn 39058  LKerclk 39086  LHypclh 39986  DVecHcdvh 41080  ocHcoch 41349  mapdcmpd 41626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-mapd 41627
This theorem is referenced by:  mapdvalc  41631  mapddlssN  41642  mapdsn  41643  mapd1o  41650  mapd0  41667
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