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Theorem lcdval 39036
 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 39035 . . . . 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 6657 . . . 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, 5syl5eq 2845 . . 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 6655 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 lcdval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8eqtr4di 2851 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
109fveq2d 6659 . . . . . 6 (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = (LDual‘𝑈))
11 lcdval.d . . . . . 6 𝐷 = (LDual‘𝑈)
1210, 11eqtr4di 2851 . . . . 5 (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = 𝐷)
139fveq2d 6659 . . . . . . 7 (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = (LFnl‘𝑈))
14 lcdval.f . . . . . . 7 𝐹 = (LFnl‘𝑈)
1513, 14eqtr4di 2851 . . . . . 6 (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = 𝐹)
16 fveq2 6655 . . . . . . . . 9 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
17 lcdval.o . . . . . . . . 9 = ((ocH‘𝐾)‘𝑊)
1816, 17eqtr4di 2851 . . . . . . . 8 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = )
199fveq2d 6659 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = (LKer‘𝑈))
20 lcdval.l . . . . . . . . . . 11 𝐿 = (LKer‘𝑈)
2119, 20eqtr4di 2851 . . . . . . . . . 10 (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = 𝐿)
2221fveq1d 6657 . . . . . . . . 9 (𝑤 = 𝑊 → ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) = (𝐿𝑓))
2318, 22fveq12d 6662 . . . . . . . 8 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) = ( ‘(𝐿𝑓)))
2418, 23fveq12d 6662 . . . . . . 7 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ( ‘( ‘(𝐿𝑓))))
2524, 22eqeq12d 2814 . . . . . 6 (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ↔ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)))
2615, 25rabeqbidv 3434 . . . . 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 7163 . . . 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 2798 . . . 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 7178 . . . 4 (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}) ∈ V
3027, 28, 29fvmpt 6755 . . 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 2853 . 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 399   = wceq 1538   ∈ wcel 2111  {crab 3110   ↦ cmpt 5114  ‘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:  lcdval2  39037  lcdlvec  39038  lcdvadd  39044  lcdsca  39046  lcdvs  39050  lcd0v  39058  lcdlsp  39068
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