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Theorem lcdval 40460
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 40459 . . . . 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 6894 . . . 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 2785 . . 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 6892 . . . . . . . 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 2791 . . . . . . 7 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = π‘ˆ)
109fveq2d 6896 . . . . . 6 (𝑀 = π‘Š β†’ (LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LDualβ€˜π‘ˆ))
11 lcdval.d . . . . . 6 𝐷 = (LDualβ€˜π‘ˆ)
1210, 11eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ (LDualβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝐷)
139fveq2d 6896 . . . . . . 7 (𝑀 = π‘Š β†’ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LFnlβ€˜π‘ˆ))
14 lcdval.f . . . . . . 7 𝐹 = (LFnlβ€˜π‘ˆ)
1513, 14eqtr4di 2791 . . . . . 6 (𝑀 = π‘Š β†’ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝐹)
16 fveq2 6892 . . . . . . . . 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 2791 . . . . . . . 8 (𝑀 = π‘Š β†’ ((ocHβ€˜πΎ)β€˜π‘€) = βŠ₯ )
199fveq2d 6896 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LKerβ€˜π‘ˆ))
20 lcdval.l . . . . . . . . . . 11 𝐿 = (LKerβ€˜π‘ˆ)
2119, 20eqtr4di 2791 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝐿)
2221fveq1d 6894 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“) = (πΏβ€˜π‘“))
2318, 22fveq12d 6899 . . . . . . . 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 6899 . . . . . . 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 2749 . . . . . 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 3450 . . . . 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 7427 . . . 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 2733 . . . 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 7442 . . . 4 (𝐷 β†Ύs {𝑓 ∈ 𝐹 ∣ ( βŠ₯ β€˜( βŠ₯ β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)}) ∈ V
3027, 28, 29fvmpt 6999 . . 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 2793 . 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 397   = wceq 1542   ∈ wcel 2107  {crab 3433   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409   β†Ύs cress 17173  LFnlclfn 37927  LKerclk 37955  LDualcld 37993  LHypclh 38855  DVecHcdvh 39949  ocHcoch 40218  LCDualclcd 40457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-lcdual 40458
This theorem is referenced by:  lcdval2  40461  lcdlvec  40462  lcdvadd  40468  lcdsca  40470  lcdvs  40474  lcd0v  40482  lcdlsp  40492
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