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Theorem lcdval 35896
 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 35895 . . . . 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 6105 . . . 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 2656 . . 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 6103 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 lcdval.u . . . . . . . 8 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
109fveq2d 6107 . . . . . 6 (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = (LDual‘𝑈))
11 lcdval.d . . . . . 6 𝐷 = (LDual‘𝑈)
1210, 11syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (LDual‘((DVecH‘𝐾)‘𝑤)) = 𝐷)
139fveq2d 6107 . . . . . . 7 (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = (LFnl‘𝑈))
14 lcdval.f . . . . . . 7 𝐹 = (LFnl‘𝑈)
1513, 14syl6eqr 2662 . . . . . 6 (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = 𝐹)
16 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
17 lcdval.o . . . . . . . . 9 = ((ocH‘𝐾)‘𝑊)
1816, 17syl6eqr 2662 . . . . . . . 8 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = )
199fveq2d 6107 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = (LKer‘𝑈))
20 lcdval.l . . . . . . . . . . 11 𝐿 = (LKer‘𝑈)
2119, 20syl6eqr 2662 . . . . . . . . . 10 (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = 𝐿)
2221fveq1d 6105 . . . . . . . . 9 (𝑤 = 𝑊 → ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) = (𝐿𝑓))
2318, 22fveq12d 6109 . . . . . . . 8 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) = ( ‘(𝐿𝑓)))
2418, 23fveq12d 6109 . . . . . . 7 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ( ‘( ‘(𝐿𝑓))))
2524, 22eqeq12d 2625 . . . . . 6 (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ↔ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)))
2615, 25rabeqbidv 3168 . . . . 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 6567 . . . 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 2610 . . . 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 6577 . . . 4 (𝐷s {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}) ∈ V
3027, 28, 29fvmpt 6191 . . 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 2664 . 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 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↾s cress 15696  LFnlclfn 33362  LKerclk 33390  LDualcld 33428  LHypclh 34288  DVecHcdvh 35385  ocHcoch 35654  LCDualclcd 35893 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-lcdual 35894 This theorem is referenced by:  lcdval2  35897  lcdlvec  35898  lcdvadd  35904  lcdsca  35906  lcdvs  35910  lcd0v  35918  lcdlsp  35928
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