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Theorem lcdval2 41496
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 41495 . 2 (𝜑𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
10 lcdval2.b . . 3 𝐵 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
1110oveq2i 7456 . 2 (𝐷s 𝐵) = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)})
129, 11eqtr4di 2792 1 (𝜑𝐶 = (𝐷s 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2103  {crab 3438  cfv 6572  (class class class)co 7445  s cress 17282  LFnlclfn 38962  LKerclk 38990  LDualcld 39028  LHypclh 39890  DVecHcdvh 40984  ocHcoch 41253  LCDualclcd 41492
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pr 5450
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-lcdual 41493
This theorem is referenced by:  lcdvbase  41499  lcdlss  41525
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