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Theorem lcdval2 41547
Description: Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
Hypotheses
Ref Expression
lcdval.h 𝐻 = (LHyp‘𝐾)
lcdval.o = ((ocH‘𝐾)‘𝑊)
lcdval.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
lcdval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcdval.f 𝐹 = (LFnl‘𝑈)
lcdval.l 𝐿 = (LKer‘𝑈)
lcdval.d 𝐷 = (LDual‘𝑈)
lcdval.k (𝜑 → (𝐾𝑋𝑊𝐻))
lcdval2.b 𝐵 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
Assertion
Ref Expression
lcdval2 (𝜑𝐶 = (𝐷s 𝐵))
Distinct variable groups:   𝑓,𝐾   𝑓,𝐹   𝑓,𝑊
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑈(𝑓)   𝐻(𝑓)   𝐿(𝑓)   (𝑓)   𝑋(𝑓)

Proof of Theorem lcdval2
StepHypRef Expression
1 lcdval.h . . 3 𝐻 = (LHyp‘𝐾)
2 lcdval.o . . 3 = ((ocH‘𝐾)‘𝑊)
3 lcdval.c . . 3 𝐶 = ((LCDual‘𝐾)‘𝑊)
4 lcdval.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
5 lcdval.f . . 3 𝐹 = (LFnl‘𝑈)
6 lcdval.l . . 3 𝐿 = (LKer‘𝑈)
7 lcdval.d . . 3 𝐷 = (LDual‘𝑈)
8 lcdval.k . . 3 (𝜑 → (𝐾𝑋𝑊𝐻))
91, 2, 3, 4, 5, 6, 7, 8lcdval 41546 . 2 (𝜑𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
10 lcdval2.b . . 3 𝐵 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
1110oveq2i 7459 . 2 (𝐷s 𝐵) = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)})
129, 11eqtr4di 2798 1 (𝜑𝐶 = (𝐷s 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  {crab 3443  cfv 6573  (class class class)co 7448  s cress 17287  LFnlclfn 39013  LKerclk 39041  LDualcld 39079  LHypclh 39941  DVecHcdvh 41035  ocHcoch 41304  LCDualclcd 41543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-lcdual 41544
This theorem is referenced by:  lcdvbase  41550  lcdlss  41576
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