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Theorem pjdm 20245
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 6344 . . . . 5 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 6823 . . . 4 (𝑥 = 𝑇 → (𝑥𝑃( 𝑥)) = (𝑇𝑃( 𝑇)))
43eleq1d 2816 . . 3 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)) ∈ (𝑉𝑚 𝑉)))
5 pjfval.v . . . . 5 𝑉 = (Base‘𝑊)
6 fvex 6354 . . . . 5 (Base‘𝑊) ∈ V
75, 6eqeltri 2827 . . . 4 𝑉 ∈ V
87, 7elmap 8044 . . 3 ((𝑇𝑃( 𝑇)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉)
94, 8syl6bb 276 . 2 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉))
10 cnvin 5690 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
11 cnvxp 5701 . . . . . . . 8 (V × (𝑉𝑚 𝑉)) = ((𝑉𝑚 𝑉) × V)
1211ineq2i 3946 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉𝑚 𝑉) × V))
1310, 12eqtri 2774 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉𝑚 𝑉) × V))
14 pjfval.l . . . . . . . 8 𝐿 = (LSubSp‘𝑊)
15 pjfval.o . . . . . . . 8 = (ocv‘𝑊)
16 pjfval.p . . . . . . . 8 𝑃 = (proj1𝑊)
17 pjfval.k . . . . . . . 8 𝐾 = (proj‘𝑊)
185, 14, 15, 16, 17pjfval 20244 . . . . . . 7 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
1918cnveqi 5444 . . . . . 6 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
20 df-res 5270 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉)) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉𝑚 𝑉) × V))
2113, 19, 203eqtr4i 2784 . . . . 5 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
2221rneqi 5499 . . . 4 ran 𝐾 = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
23 dfdm4 5463 . . . 4 dom 𝐾 = ran 𝐾
24 df-ima 5271 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉)) = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
2522, 23, 243eqtr4i 2784 . . 3 dom 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉))
26 eqid 2752 . . . 4 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
2726mptpreima 5781 . . 3 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉)) = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉)}
2825, 27eqtri 2774 . 2 dom 𝐾 = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉)}
299, 28elrab2 3499 1 (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1624  wcel 2131  {crab 3046  Vcvv 3332  cin 3706  cmpt 4873   × cxp 5256  ccnv 5257  dom cdm 5258  ran crn 5259  cres 5260  cima 5261  wf 6037  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  Basecbs 16051  proj1cpj1 18242  LSubSpclss 19126  ocvcocv 20198  projcpj 20238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-map 8017  df-pj 20241
This theorem is referenced by:  pjfval2  20247  pjdm2  20249  pjf  20251
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