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Theorem pjdm 21246
Description: A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval.v 𝑉 = (Base‘𝑊)
pjfval.l 𝐿 = (LSubSp‘𝑊)
pjfval.o = (ocv‘𝑊)
pjfval.p 𝑃 = (proj1𝑊)
pjfval.k 𝐾 = (proj‘𝑊)
Assertion
Ref Expression
pjdm (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))

Proof of Theorem pjdm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (𝑥 = 𝑇𝑥 = 𝑇)
2 fveq2 6888 . . . . 5 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 7422 . . . 4 (𝑥 = 𝑇 → (𝑥𝑃( 𝑥)) = (𝑇𝑃( 𝑇)))
43eleq1d 2819 . . 3 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)) ∈ (𝑉m 𝑉)))
5 pjfval.v . . . . 5 𝑉 = (Base‘𝑊)
65fvexi 6902 . . . 4 𝑉 ∈ V
76, 6elmap 8861 . . 3 ((𝑇𝑃( 𝑇)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉)
84, 7bitrdi 287 . 2 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉))
9 cnvin 6141 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
10 cnvxp 6153 . . . . . . . 8 (V × (𝑉m 𝑉)) = ((𝑉m 𝑉) × V)
1110ineq2i 4208 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉m 𝑉) × V))
129, 11eqtri 2761 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉m 𝑉) × V))
13 pjfval.l . . . . . . . 8 𝐿 = (LSubSp‘𝑊)
14 pjfval.o . . . . . . . 8 = (ocv‘𝑊)
15 pjfval.p . . . . . . . 8 𝑃 = (proj1𝑊)
16 pjfval.k . . . . . . . 8 𝐾 = (proj‘𝑊)
175, 13, 14, 15, 16pjfval 21245 . . . . . . 7 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
1817cnveqi 5872 . . . . . 6 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
19 df-res 5687 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉)) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉m 𝑉) × V))
2012, 18, 193eqtr4i 2771 . . . . 5 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
2120rneqi 5934 . . . 4 ran 𝐾 = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
22 dfdm4 5893 . . . 4 dom 𝐾 = ran 𝐾
23 df-ima 5688 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉)) = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
2421, 22, 233eqtr4i 2771 . . 3 dom 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉))
25 eqid 2733 . . . 4 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
2625mptpreima 6234 . . 3 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉)) = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉)}
2724, 26eqtri 2761 . 2 dom 𝐾 = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉)}
288, 27elrab2 3685 1 (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cin 3946  cmpt 5230   × cxp 5673  ccnv 5674  dom cdm 5675  ran crn 5676  cres 5677  cima 5678  wf 6536  cfv 6540  (class class class)co 7404  m cmap 8816  Basecbs 17140  proj1cpj1 19496  LSubSpclss 20530  ocvcocv 21197  projcpj 21239
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-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
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-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8818  df-pj 21242
This theorem is referenced by:  pjfval2  21248  pjdm2  21250  pjf  21252
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