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Theorem lcdval2 39037
 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 39036 . 2 (𝜑𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
10 lcdval2.b . . 3 𝐵 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
1110oveq2i 7156 . 2 (𝐷s 𝐵) = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)})
129, 11eqtr4di 2851 1 (𝜑𝐶 = (𝐷s 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  ‘cfv 6332  (class class class)co 7145   ↾s cress 16496  LFnlclfn 36504  LKerclk 36532  LDualcld 36570  LHypclh 37431  DVecHcdvh 38525  ocHcoch 38794  LCDualclcd 39033 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-lcdual 39034 This theorem is referenced by:  lcdvbase  39040  lcdlss  39066
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