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Theorem mapdvalc 38645
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)
mapdvalc.c 𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}
Assertion
Ref Expression
mapdvalc (𝜑 → (𝑀𝑇) = {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑇})
Distinct variable groups:   𝑓,𝐾   𝑓,𝐹   𝑓,𝑊   𝑓,𝑔,𝐹   𝑔,𝐿   𝑔,𝑂   𝑇,𝑓   𝜑,𝑓
Allowed substitution hints:   𝜑(𝑔)   𝐶(𝑓,𝑔)   𝑆(𝑓,𝑔)   𝑇(𝑔)   𝑈(𝑓,𝑔)   𝐻(𝑓,𝑔)   𝐾(𝑔)   𝐿(𝑓)   𝑀(𝑓,𝑔)   𝑂(𝑓)   𝑊(𝑔)   𝑋(𝑓,𝑔)

Proof of Theorem mapdvalc
StepHypRef Expression
1 mapdval.h . . 3 𝐻 = (LHyp‘𝐾)
2 mapdval.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 mapdval.s . . 3 𝑆 = (LSubSp‘𝑈)
4 mapdval.f . . 3 𝐹 = (LFnl‘𝑈)
5 mapdval.l . . 3 𝐿 = (LKer‘𝑈)
6 mapdval.o . . 3 𝑂 = ((ocH‘𝐾)‘𝑊)
7 mapdval.m . . 3 𝑀 = ((mapd‘𝐾)‘𝑊)
8 mapdval.k . . 3 (𝜑 → (𝐾𝑋𝑊𝐻))
9 mapdval.t . . 3 (𝜑𝑇𝑆)
101, 2, 3, 4, 5, 6, 7, 8, 9mapdval 38644 . 2 (𝜑 → (𝑀𝑇) = {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)})
11 anass 469 . . . 4 (((𝑓𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇) ↔ (𝑓𝐹 ∧ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)))
12 mapdvalc.c . . . . . . . 8 𝐶 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) = (𝐿𝑔)}
1312lcfl1lem 38507 . . . . . . 7 (𝑓𝐶 ↔ (𝑓𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)))
1413anbi1i 623 . . . . . 6 ((𝑓𝐶 ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇) ↔ ((𝑓𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇))
1514bicomi 225 . . . . 5 (((𝑓𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇) ↔ (𝑓𝐶 ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇))
1615a1i 11 . . . 4 (𝜑 → (((𝑓𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓)) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇) ↔ (𝑓𝐶 ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)))
1711, 16syl5bbr 286 . . 3 (𝜑 → ((𝑓𝐹 ∧ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)) ↔ (𝑓𝐶 ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)))
1817rabbidva2 3474 . 2 (𝜑 → {𝑓𝐹 ∣ ((𝑂‘(𝑂‘(𝐿𝑓))) = (𝐿𝑓) ∧ (𝑂‘(𝐿𝑓)) ⊆ 𝑇)} = {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑇})
1910, 18eqtrd 2853 1 (𝜑 → (𝑀𝑇) = {𝑓𝐶 ∣ (𝑂‘(𝐿𝑓)) ⊆ 𝑇})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  wss 3933  cfv 6348  LSubSpclss 19632  LFnlclfn 36073  LKerclk 36101  LHypclh 37000  DVecHcdvh 38094  ocHcoch 38363  mapdcmpd 38640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-mapd 38641
This theorem is referenced by:  mapdval2N  38646  mapdordlem2  38653  mapdrval  38663
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