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Theorem pjdm 21727
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 6906 . . . . 5 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 7449 . . . 4 (𝑥 = 𝑇 → (𝑥𝑃( 𝑥)) = (𝑇𝑃( 𝑇)))
43eleq1d 2826 . . 3 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)) ∈ (𝑉m 𝑉)))
5 pjfval.v . . . . 5 𝑉 = (Base‘𝑊)
65fvexi 6920 . . . 4 𝑉 ∈ V
76, 6elmap 8911 . . 3 ((𝑇𝑃( 𝑇)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉)
84, 7bitrdi 287 . 2 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉))
9 cnvin 6164 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
10 cnvxp 6177 . . . . . . . 8 (V × (𝑉m 𝑉)) = ((𝑉m 𝑉) × V)
1110ineq2i 4217 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉m 𝑉) × V))
129, 11eqtri 2765 . . . . . 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 21726 . . . . . . 7 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
1817cnveqi 5885 . . . . . 6 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
19 df-res 5697 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉)) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉m 𝑉) × V))
2012, 18, 193eqtr4i 2775 . . . . 5 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
2120rneqi 5948 . . . 4 ran 𝐾 = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
22 dfdm4 5906 . . . 4 dom 𝐾 = ran 𝐾
23 df-ima 5698 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉)) = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
2421, 22, 233eqtr4i 2775 . . 3 dom 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉))
25 eqid 2737 . . . 4 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
2625mptpreima 6258 . . 3 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉)) = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉)}
2724, 26eqtri 2765 . 2 dom 𝐾 = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉)}
288, 27elrab2 3695 1 (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cin 3950  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  Basecbs 17247  proj1cpj1 19653  LSubSpclss 20929  ocvcocv 21678  projcpj 21720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-pj 21723
This theorem is referenced by:  pjfval2  21729  pjdm2  21731  pjf  21733
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