Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcdval2 Structured version   Visualization version   GIF version

Theorem lcdval2 38606
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 38605 . 2 (𝜑𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
10 lcdval2.b . . 3 𝐵 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
1110oveq2i 7156 . 2 (𝐷s 𝐵) = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)})
129, 11syl6eqr 2871 1 (𝜑𝐶 = (𝐷s 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {crab 3139  cfv 6348  (class class class)co 7145  s cress 16472  LFnlclfn 36073  LKerclk 36101  LDualcld 36139  LHypclh 37000  DVecHcdvh 38094  ocHcoch 38363  LCDualclcd 38602
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-ov 7148  df-lcdual 38603
This theorem is referenced by:  lcdvbase  38609  lcdlss  38635
  Copyright terms: Public domain W3C validator