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Theorem pjdm 20405
 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 6650 . . . . 5 (𝑥 = 𝑇 → ( 𝑥) = ( 𝑇))
31, 2oveq12d 7158 . . . 4 (𝑥 = 𝑇 → (𝑥𝑃( 𝑥)) = (𝑇𝑃( 𝑇)))
43eleq1d 2874 . . 3 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)) ∈ (𝑉m 𝑉)))
5 pjfval.v . . . . 5 𝑉 = (Base‘𝑊)
65fvexi 6664 . . . 4 𝑉 ∈ V
76, 6elmap 8425 . . 3 ((𝑇𝑃( 𝑇)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉)
84, 7syl6bb 290 . 2 (𝑥 = 𝑇 → ((𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉) ↔ (𝑇𝑃( 𝑇)):𝑉𝑉))
9 cnvin 5971 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
10 cnvxp 5982 . . . . . . . 8 (V × (𝑉m 𝑉)) = ((𝑉m 𝑉) × V)
1110ineq2i 4136 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉m 𝑉) × V))
129, 11eqtri 2821 . . . . . 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 20404 . . . . . . 7 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
1817cnveqi 5710 . . . . . 6 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
19 df-res 5532 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉)) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ ((𝑉m 𝑉) × V))
2012, 18, 193eqtr4i 2831 . . . . 5 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
2120rneqi 5772 . . . 4 ran 𝐾 = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
22 dfdm4 5729 . . . 4 dom 𝐾 = ran 𝐾
23 df-ima 5533 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉)) = ran ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ↾ (𝑉m 𝑉))
2421, 22, 233eqtr4i 2831 . . 3 dom 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉))
25 eqid 2798 . . . 4 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
2625mptpreima 6060 . . 3 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) “ (𝑉m 𝑉)) = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉)}
2724, 26eqtri 2821 . 2 dom 𝐾 = {𝑥𝐿 ∣ (𝑥𝑃( 𝑥)) ∈ (𝑉m 𝑉)}
288, 27elrab2 3631 1 (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ∩ cin 3880   ↦ cmpt 5111   × cxp 5518  ◡ccnv 5519  dom cdm 5520  ran crn 5521   ↾ cres 5522   “ cima 5523  ⟶wf 6323  ‘cfv 6327  (class class class)co 7140   ↑m cmap 8396  Basecbs 16482  proj1cpj1 18760  LSubSpclss 19704  ocvcocv 20358  projcpj 20398 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8398  df-pj 20401 This theorem is referenced by:  pjfval2  20407  pjdm2  20409  pjf  20411
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