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Theorem pjdm 19983
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 6153 . . . . 5 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 6628 . . . 4 (𝑥 = 𝑇 → (𝑥𝑃( 𝑥)) = (𝑇𝑃( 𝑇)))
43eleq1d 2683 . . 3 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)) ∈ (𝑉𝑚 𝑉)))
5 pjfval.v . . . . 5 𝑉 = (Base‘𝑊)
6 fvex 6163 . . . . 5 (Base‘𝑊) ∈ V
75, 6eqeltri 2694 . . . 4 𝑉 ∈ V
87, 7elmap 7838 . . 3 ((𝑇𝑃( 𝑇)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉)
94, 8syl6bb 276 . 2 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉))
10 cnvin 5504 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
11 cnvxp 5515 . . . . . . . 8 (V × (𝑉𝑚 𝑉)) = ((𝑉𝑚 𝑉) × V)
1211ineq2i 3794 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉𝑚 𝑉) × V))
1310, 12eqtri 2643 . . . . . 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 19982 . . . . . . 7 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
1918cnveqi 5262 . . . . . 6 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
20 df-res 5091 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉)) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉𝑚 𝑉) × V))
2113, 19, 203eqtr4i 2653 . . . . 5 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
2221rneqi 5317 . . . 4 ran 𝐾 = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
23 dfdm4 5281 . . . 4 dom 𝐾 = ran 𝐾
24 df-ima 5092 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉)) = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉𝑚 𝑉))
2522, 23, 243eqtr4i 2653 . . 3 dom 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉))
26 eqid 2621 . . . 4 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
2726mptpreima 5592 . . 3 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉𝑚 𝑉)) = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉)}
2825, 27eqtri 2643 . 2 dom 𝐾 = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉𝑚 𝑉)}
299, 28elrab2 3352 1 (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3189  cin 3558  cmpt 4678   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  Basecbs 15792  proj1cpj1 17982  LSubSpclss 18864  ocvcocv 19936  projcpj 19976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-pj 19979
This theorem is referenced by:  pjfval2  19985  pjdm2  19987  pjf  19989
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