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Theorem mapdfval 41011
Description: Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)
Hypotheses
Ref Expression
mapdval.h 𝐻 = (LHypβ€˜πΎ)
mapdval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
mapdval.s 𝑆 = (LSubSpβ€˜π‘ˆ)
mapdval.f 𝐹 = (LFnlβ€˜π‘ˆ)
mapdval.l 𝐿 = (LKerβ€˜π‘ˆ)
mapdval.o 𝑂 = ((ocHβ€˜πΎ)β€˜π‘Š)
mapdval.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
mapdfval ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)}))
Distinct variable groups:   𝑓,𝑠,𝐾   𝑓,𝐹   𝑆,𝑠   𝑓,π‘Š,𝑠
Allowed substitution hints:   𝑆(𝑓)   π‘ˆ(𝑓,𝑠)   𝐹(𝑠)   𝐻(𝑓,𝑠)   𝐿(𝑓,𝑠)   𝑀(𝑓,𝑠)   𝑂(𝑓,𝑠)   𝑋(𝑓,𝑠)

Proof of Theorem mapdfval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 mapdval.m . . 3 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
2 mapdval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
32mapdffval 41010 . . . 4 (𝐾 ∈ 𝑋 β†’ (mapdβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ {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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)})))
43fveq1d 6887 . . 3 (𝐾 ∈ 𝑋 β†’ ((mapdβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ {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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)}))β€˜π‘Š))
51, 4eqtrid 2778 . 2 (𝐾 ∈ 𝑋 β†’ 𝑀 = ((𝑀 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ {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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)}))β€˜π‘Š))
6 fveq2 6885 . . . . . . 7 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘Š))
7 mapdval.u . . . . . . 7 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
86, 7eqtr4di 2784 . . . . . 6 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = π‘ˆ)
98fveq2d 6889 . . . . 5 (𝑀 = π‘Š β†’ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LSubSpβ€˜π‘ˆ))
10 mapdval.s . . . . 5 𝑆 = (LSubSpβ€˜π‘ˆ)
119, 10eqtr4di 2784 . . . 4 (𝑀 = π‘Š β†’ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝑆)
128fveq2d 6889 . . . . . 6 (𝑀 = π‘Š β†’ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LFnlβ€˜π‘ˆ))
13 mapdval.f . . . . . 6 𝐹 = (LFnlβ€˜π‘ˆ)
1412, 13eqtr4di 2784 . . . . 5 (𝑀 = π‘Š β†’ (LFnlβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝐹)
15 fveq2 6885 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((ocHβ€˜πΎ)β€˜π‘€) = ((ocHβ€˜πΎ)β€˜π‘Š))
16 mapdval.o . . . . . . . . 9 𝑂 = ((ocHβ€˜πΎ)β€˜π‘Š)
1715, 16eqtr4di 2784 . . . . . . . 8 (𝑀 = π‘Š β†’ ((ocHβ€˜πΎ)β€˜π‘€) = 𝑂)
188fveq2d 6889 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (LKerβ€˜π‘ˆ))
19 mapdval.l . . . . . . . . . . 11 𝐿 = (LKerβ€˜π‘ˆ)
2018, 19eqtr4di 2784 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝐿)
2120fveq1d 6887 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“) = (πΏβ€˜π‘“))
2217, 21fveq12d 6892 . . . . . . . 8 (𝑀 = π‘Š β†’ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) = (π‘‚β€˜(πΏβ€˜π‘“)))
2317, 22fveq12d 6892 . . . . . . 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‘“))) = (π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))))
2423, 21eqeq12d 2742 . . . . . 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‘“) ↔ (π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“)))
2522sseq1d 4008 . . . . . 6 (𝑀 = π‘Š β†’ ((((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠 ↔ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠))
2624, 25anbi12d 630 . . . . 5 (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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠) ↔ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)))
2714, 26rabeqbidv 3443 . . . 4 (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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)} = {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)})
2811, 27mpteq12dv 5232 . . 3 (𝑀 = π‘Š β†’ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ {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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)}) = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)}))
29 eqid 2726 . . 3 (𝑀 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ {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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)})) = (𝑀 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ {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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)}))
3028, 29, 10mptfvmpt 7225 . 2 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSpβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ {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‘“) ∧ (((ocHβ€˜πΎ)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜πΎ)β€˜π‘€))β€˜π‘“)) βŠ† 𝑠)}))β€˜π‘Š) = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)}))
315, 30sylan9eq 2786 1 ((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) β†’ 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426   βŠ† wss 3943   ↦ cmpt 5224  β€˜cfv 6537  LSubSpclss 20778  LFnlclfn 38440  LKerclk 38468  LHypclh 39368  DVecHcdvh 40462  ocHcoch 40731  mapdcmpd 41008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-mapd 41009
This theorem is referenced by:  mapdval  41012  mapd1o  41032
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