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Theorem mapdval 40120
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 40119 . . . 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 6849 . 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 6861 . . . 4 𝐹 ∈ V
1413rabex 5294 . . 3 {𝑓 ∈ 𝐹 ∣ ((π‘‚β€˜(π‘‚β€˜(πΏβ€˜π‘“))) = (πΏβ€˜π‘“) ∧ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇)} ∈ V
15 sseq2 3975 . . . . . 6 (𝑠 = 𝑇 β†’ ((π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑠 ↔ (π‘‚β€˜(πΏβ€˜π‘“)) βŠ† 𝑇))
1615anbi2d 630 . . . . 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 3418 . . . 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 2737 . . . 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 6951 . . 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 587 . 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 2777 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 397   = wceq 1542   ∈ wcel 2107  {crab 3410  Vcvv 3448   βŠ† wss 3915   ↦ cmpt 5193  β€˜cfv 6501  LSubSpclss 20408  LFnlclfn 37548  LKerclk 37576  LHypclh 38476  DVecHcdvh 39570  ocHcoch 39839  mapdcmpd 40116
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-mapd 40117
This theorem is referenced by:  mapdvalc  40121  mapddlssN  40132  mapdsn  40133  mapd1o  40140  mapd0  40157
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