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Theorem lcdval 41546
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))
Assertion
Ref Expression
lcdval (𝜑𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
Distinct variable groups:   𝑓,𝐾   𝑓,𝐹   𝑓,𝑊
Allowed substitution hints:   𝜑(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑈(𝑓)   𝐻(𝑓)   𝐿(𝑓)   (𝑓)   𝑋(𝑓)

Proof of Theorem lcdval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lcdval.k . 2 (𝜑 → (𝐾𝑋𝑊𝐻))
2 lcdval.c . . . 4 𝐶 = ((LCDual‘𝐾)‘𝑊)
3 lcdval.h . . . . . 6 𝐻 = (LHyp‘𝐾)
43lcdfval 41545 . . . . 5 (𝐾𝑋 → (LCDual‘𝐾) = (𝑤𝐻 ↦ ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)})))
54fveq1d 6922 . . . 4 (𝐾𝑋 → ((LCDual‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)}))‘𝑊))
62, 5eqtrid 2792 . . 3 (𝐾𝑋𝐶 = ((𝑤𝐻 ↦ ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)}))‘𝑊))
7 fveq2 6920 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 lcdval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8eqtr4di 2798 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
109fveq2d 6924 . . . . . 6 (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = (LDual‘𝑈))
11 lcdval.d . . . . . 6 𝐷 = (LDual‘𝑈)
1210, 11eqtr4di 2798 . . . . 5 (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = 𝐷)
139fveq2d 6924 . . . . . . 7 (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = (LFnl‘𝑈))
14 lcdval.f . . . . . . 7 𝐹 = (LFnl‘𝑈)
1513, 14eqtr4di 2798 . . . . . 6 (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = 𝐹)
16 fveq2 6920 . . . . . . . . 9 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
17 lcdval.o . . . . . . . . 9 = ((ocH‘𝐾)‘𝑊)
1816, 17eqtr4di 2798 . . . . . . . 8 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = )
199fveq2d 6924 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = (LKer‘𝑈))
20 lcdval.l . . . . . . . . . . 11 𝐿 = (LKer‘𝑈)
2119, 20eqtr4di 2798 . . . . . . . . . 10 (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = 𝐿)
2221fveq1d 6922 . . . . . . . . 9 (𝑤 = 𝑊 → ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) = (𝐿𝑓))
2318, 22fveq12d 6927 . . . . . . . 8 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) = ( ‘(𝐿𝑓)))
2418, 23fveq12d 6927 . . . . . . 7 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ( ‘( ‘(𝐿𝑓))))
2524, 22eqeq12d 2756 . . . . . 6 (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ↔ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)))
2615, 25rabeqbidv 3462 . . . . 5 (𝑤 = 𝑊 → {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)} = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)})
2712, 26oveq12d 7466 . . . 4 (𝑤 = 𝑊 → ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)}) = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
28 eqid 2740 . . . 4 (𝑤𝐻 ↦ ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)})) = (𝑤𝐻 ↦ ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)}))
29 ovex 7481 . . . 4 (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}) ∈ V
3027, 28, 29fvmpt 7029 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ ((LDual‘((DVecH‘𝐾)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)}))‘𝑊) = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
316, 30sylan9eq 2800 . 2 ((𝐾𝑋𝑊𝐻) → 𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
321, 31syl 17 1 (𝜑𝐶 = (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  {crab 3443  cmpt 5249  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:  lcdval2  41547  lcdlvec  41548  lcdvadd  41554  lcdsca  41556  lcdvs  41560  lcd0v  41568  lcdlsp  41578
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