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Theorem lcdval 40924
Description: Dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.)
Hypotheses
Ref Expression
lcdval.h 𝐻 = (LHypβ€˜πΎ)
lcdval.o βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)
lcdval.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
lcdval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
lcdval.f 𝐹 = (LFnlβ€˜π‘ˆ)
lcdval.l 𝐿 = (LKerβ€˜π‘ˆ)
lcdval.d 𝐷 = (LDualβ€˜π‘ˆ)
lcdval.k (πœ‘ β†’ (𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻))
Assertion
Ref Expression
lcdval (πœ‘ β†’ 𝐢 = (𝐷 β†Ύs {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}))
Distinct variable groups:   𝑓,𝐾   𝑓,𝐹   𝑓,π‘Š
Allowed substitution hints:   πœ‘(𝑓)   𝐢(𝑓)   𝐷(𝑓)   π‘ˆ(𝑓)   𝐻(𝑓)   𝐿(𝑓)   βŠ₯ (𝑓)   𝑋(𝑓)

Proof of Theorem lcdval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 lcdval.k . 2 (πœ‘ β†’ (𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻))
2 lcdval.c . . . 4 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
3 lcdval.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
43lcdfval 40923 . . . . 5 (𝐾 ∈ 𝑋 β†’ (LCDualβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ ((LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)})))
54fveq1d 6893 . . . 4 (𝐾 ∈ 𝑋 β†’ ((LCDualβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ ((LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)}))β€˜π‘Š))
62, 5eqtrid 2783 . . 3 (𝐾 ∈ 𝑋 β†’ 𝐢 = ((𝑀 ∈ 𝐻 ↦ ((LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)}))β€˜π‘Š))
7 fveq2 6891 . . . . . . . 8 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘Š))
8 lcdval.u . . . . . . . 8 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
97, 8eqtr4di 2789 . . . . . . 7 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = π‘ˆ)
109fveq2d 6895 . . . . . 6 (𝑀 = π‘Š β†’ (LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LDualβ€˜π‘ˆ))
11 lcdval.d . . . . . 6 𝐷 = (LDualβ€˜π‘ˆ)
1210, 11eqtr4di 2789 . . . . 5 (𝑀 = π‘Š β†’ (LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝐷)
139fveq2d 6895 . . . . . . 7 (𝑀 = π‘Š β†’ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LFnlβ€˜π‘ˆ))
14 lcdval.f . . . . . . 7 𝐹 = (LFnlβ€˜π‘ˆ)
1513, 14eqtr4di 2789 . . . . . 6 (𝑀 = π‘Š β†’ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝐹)
16 fveq2 6891 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((ocHβ€˜πΎ)β€˜π‘€) = ((ocHβ€˜πΎ)β€˜π‘Š))
17 lcdval.o . . . . . . . . 9 βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)
1816, 17eqtr4di 2789 . . . . . . . 8 (𝑀 = π‘Š β†’ ((ocHβ€˜πΎ)β€˜π‘€) = βŠ₯ )
199fveq2d 6895 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LKerβ€˜π‘ˆ))
20 lcdval.l . . . . . . . . . . 11 𝐿 = (LKerβ€˜π‘ˆ)
2119, 20eqtr4di 2789 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝐿)
2221fveq1d 6893 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“) = (πΏβ€˜π‘“))
2318, 22fveq12d 6898 . . . . . . . 8 (𝑀 = π‘Š β†’ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) = ( βŠ₯ β€˜(πΏβ€˜π‘“)))
2418, 23fveq12d 6898 . . . . . . 7 (𝑀 = π‘Š β†’ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))))
2524, 22eqeq12d 2747 . . . . . 6 (𝑀 = π‘Š β†’ ((((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“) ↔ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)))
2615, 25rabeqbidv 3448 . . . . 5 (𝑀 = π‘Š β†’ {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)} = {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)})
2712, 26oveq12d 7430 . . . 4 (𝑀 = π‘Š β†’ ((LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)}) = (𝐷 β†Ύs {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}))
28 eqid 2731 . . . 4 (𝑀 ∈ 𝐻 ↦ ((LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)})) = (𝑀 ∈ 𝐻 ↦ ((LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)}))
29 ovex 7445 . . . 4 (𝐷 β†Ύs {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}) ∈ V
3027, 28, 29fvmpt 6998 . . 3 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ ((LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ∣ (((ocHβ€˜πΎ)β€˜π‘€)β€˜(((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)}))β€˜π‘Š) = (𝐷 β†Ύs {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}))
316, 30sylan9eq 2791 . 2 ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝐢 = (𝐷 β†Ύs {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}))
321, 31syl 17 1 (πœ‘ β†’ 𝐢 = (𝐷 β†Ύs {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {crab 3431   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7412   β†Ύs cress 17180  LFnlclfn 38391  LKerclk 38419  LDualcld 38457  LHypclh 39319  DVecHcdvh 40413  ocHcoch 40682  LCDualclcd 40921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-lcdual 40922
This theorem is referenced by:  lcdval2  40925  lcdlvec  40926  lcdvadd  40932  lcdsca  40934  lcdvs  40938  lcd0v  40946  lcdlsp  40956
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