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Theorem pjdm 19870
 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 6103 . . . . 5 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 6567 . . . 4 (𝑥 = 𝑇 → (𝑥𝑃( 𝑥)) = (𝑇𝑃( 𝑇)))
43eleq1d 2672 . . 3 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)) ∈ (𝑉𝑚 𝑉)))
5 pjfval.v . . . . 5 𝑉 = (Base‘𝑊)
6 fvex 6113 . . . . 5 (Base‘𝑊) ∈ V
75, 6eqeltri 2684 . . . 4 𝑉 ∈ V
87, 7elmap 7772 . . 3 ((𝑇𝑃( 𝑇)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉)
94, 8syl6bb 275 . 2 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉))
10 cnvin 5459 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
11 cnvxp 5470 . . . . . . . 8 (V × (𝑉𝑚 𝑉)) = ((𝑉𝑚 𝑉) × V)
1211ineq2i 3773 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉𝑚 𝑉) × V))
1310, 12eqtri 2632 . . . . . 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 19869 . . . . . . 7 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
1918cnveqi 5219 . . . . . 6 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
20 df-res 5050 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉)) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉𝑚 𝑉) × V))
2113, 19, 203eqtr4i 2642 . . . . 5 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
2221rneqi 5273 . . . 4 ran 𝐾 = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
23 dfdm4 5238 . . . 4 dom 𝐾 = ran 𝐾
24 df-ima 5051 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉)) = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
2522, 23, 243eqtr4i 2642 . . 3 dom 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉))
26 eqid 2610 . . . 4 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
2726mptpreima 5545 . . 3 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉)) = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉)}
2825, 27eqtri 2632 . 2 dom 𝐾 = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉)}
299, 28elrab2 3333 1 (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Basecbs 15695  proj1cpj1 17873  LSubSpclss 18753  ocvcocv 19823  projcpj 19863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-pj 19866 This theorem is referenced by:  pjfval2  19872  pjdm2  19874  pjf  19876
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