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Theorem mapdval 41012
Description: Value of projectivity from vector space H to dual space. (Contributed by NM, 27-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)
mapdval.k (πœ‘ β†’ (𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻))
mapdval.t (πœ‘ β†’ 𝑇 ∈ 𝑆)
Assertion
Ref Expression
mapdval (πœ‘ β†’ (π‘€β€˜π‘‡) = {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)})
Distinct variable groups:   𝑓,𝐾   𝑓,𝐹   𝑓,π‘Š   𝑇,𝑓
Allowed substitution hints:   πœ‘(𝑓)   𝑆(𝑓)   π‘ˆ(𝑓)   𝐻(𝑓)   𝐿(𝑓)   𝑀(𝑓)   𝑂(𝑓)   𝑋(𝑓)

Proof of Theorem mapdval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 mapdval.k . . . 4 (πœ‘ β†’ (𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻))
2 mapdval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
3 mapdval.u . . . . 5 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
4 mapdval.s . . . . 5 𝑆 = (LSubSpβ€˜π‘ˆ)
5 mapdval.f . . . . 5 𝐹 = (LFnlβ€˜π‘ˆ)
6 mapdval.l . . . . 5 𝐿 = (LKerβ€˜π‘ˆ)
7 mapdval.o . . . . 5 𝑂 = ((ocHβ€˜πΎ)β€˜π‘Š)
8 mapdval.m . . . . 5 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
92, 3, 4, 5, 6, 7, 8mapdfval 41011 . . . 4 ((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)}))
101, 9syl 17 . . 3 (πœ‘ β†’ 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)}))
1110fveq1d 6887 . 2 (πœ‘ β†’ (π‘€β€˜π‘‡) = ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)})β€˜π‘‡))
12 mapdval.t . . 3 (πœ‘ β†’ 𝑇 ∈ 𝑆)
135fvexi 6899 . . . 4 𝐹 ∈ V
1413rabex 5325 . . 3 {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)} ∈ V
15 sseq2 4003 . . . . . 6 (𝑠 = 𝑇 β†’ ((π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠 ↔ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇))
1615anbi2d 628 . . . . 5 (𝑠 = 𝑇 β†’ (((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠) ↔ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)))
1716rabbidv 3434 . . . 4 (𝑠 = 𝑇 β†’ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)} = {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)})
18 eqid 2726 . . . 4 (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)}) = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)})
1917, 18fvmptg 6990 . . 3 ((𝑇 ∈ 𝑆 ∧ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)} ∈ V) β†’ ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)})β€˜π‘‡) = {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)})
2012, 14, 19sylancl 585 . 2 (πœ‘ β†’ ((𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠)})β€˜π‘‡) = {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)})
2111, 20eqtrd 2766 1 (πœ‘ β†’ (π‘€β€˜π‘‡) = {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   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:  mapdvalc  41013  mapddlssN  41024  mapdsn  41025  mapd1o  41032  mapd0  41049
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