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Theorem pjdm 21262
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 = (Baseβ€˜π‘Š)
pjfval.l 𝐿 = (LSubSpβ€˜π‘Š)
pjfval.o βŠ₯ = (ocvβ€˜π‘Š)
pjfval.p 𝑃 = (proj1β€˜π‘Š)
pjfval.k 𝐾 = (projβ€˜π‘Š)
Assertion
Ref Expression
pjdm (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( βŠ₯ β€˜π‘‡)):π‘‰βŸΆπ‘‰))

Proof of Theorem pjdm
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (π‘₯ = 𝑇 β†’ π‘₯ = 𝑇)
2 fveq2 6892 . . . . 5 (π‘₯ = 𝑇 β†’ ( βŠ₯ β€˜π‘₯) = ( βŠ₯ β€˜π‘‡))
31, 2oveq12d 7427 . . . 4 (π‘₯ = 𝑇 β†’ (π‘₯𝑃( βŠ₯ β€˜π‘₯)) = (𝑇𝑃( βŠ₯ β€˜π‘‡)))
43eleq1d 2819 . . 3 (π‘₯ = 𝑇 β†’ ((π‘₯𝑃( βŠ₯ β€˜π‘₯)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( βŠ₯ β€˜π‘‡)) ∈ (𝑉 ↑m 𝑉)))
5 pjfval.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
65fvexi 6906 . . . 4 𝑉 ∈ V
76, 6elmap 8865 . . 3 ((𝑇𝑃( βŠ₯ β€˜π‘‡)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( βŠ₯ β€˜π‘‡)):π‘‰βŸΆπ‘‰)
84, 7bitrdi 287 . 2 (π‘₯ = 𝑇 β†’ ((π‘₯𝑃( βŠ₯ β€˜π‘₯)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( βŠ₯ β€˜π‘‡)):π‘‰βŸΆπ‘‰))
9 cnvin 6145 . . . . . . 7 β—‘((π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ (V Γ— (𝑉 ↑m 𝑉))) = (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ β—‘(V Γ— (𝑉 ↑m 𝑉)))
10 cnvxp 6157 . . . . . . . 8 β—‘(V Γ— (𝑉 ↑m 𝑉)) = ((𝑉 ↑m 𝑉) Γ— V)
1110ineq2i 4210 . . . . . . 7 (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ β—‘(V Γ— (𝑉 ↑m 𝑉))) = (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ ((𝑉 ↑m 𝑉) Γ— V))
129, 11eqtri 2761 . . . . . 6 β—‘((π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ (V Γ— (𝑉 ↑m 𝑉))) = (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ ((𝑉 ↑m 𝑉) Γ— V))
13 pjfval.l . . . . . . . 8 𝐿 = (LSubSpβ€˜π‘Š)
14 pjfval.o . . . . . . . 8 βŠ₯ = (ocvβ€˜π‘Š)
15 pjfval.p . . . . . . . 8 𝑃 = (proj1β€˜π‘Š)
16 pjfval.k . . . . . . . 8 𝐾 = (projβ€˜π‘Š)
175, 13, 14, 15, 16pjfval 21261 . . . . . . 7 𝐾 = ((π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ (V Γ— (𝑉 ↑m 𝑉)))
1817cnveqi 5875 . . . . . 6 ◑𝐾 = β—‘((π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ (V Γ— (𝑉 ↑m 𝑉)))
19 df-res 5689 . . . . . 6 (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) β†Ύ (𝑉 ↑m 𝑉)) = (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) ∩ ((𝑉 ↑m 𝑉) Γ— V))
2012, 18, 193eqtr4i 2771 . . . . 5 ◑𝐾 = (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) β†Ύ (𝑉 ↑m 𝑉))
2120rneqi 5937 . . . 4 ran ◑𝐾 = ran (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) β†Ύ (𝑉 ↑m 𝑉))
22 dfdm4 5896 . . . 4 dom 𝐾 = ran ◑𝐾
23 df-ima 5690 . . . 4 (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) β€œ (𝑉 ↑m 𝑉)) = ran (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) β†Ύ (𝑉 ↑m 𝑉))
2421, 22, 233eqtr4i 2771 . . 3 dom 𝐾 = (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) β€œ (𝑉 ↑m 𝑉))
25 eqid 2733 . . . 4 (π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) = (π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯)))
2625mptpreima 6238 . . 3 (β—‘(π‘₯ ∈ 𝐿 ↦ (π‘₯𝑃( βŠ₯ β€˜π‘₯))) β€œ (𝑉 ↑m 𝑉)) = {π‘₯ ∈ 𝐿 ∣ (π‘₯𝑃( βŠ₯ β€˜π‘₯)) ∈ (𝑉 ↑m 𝑉)}
2724, 26eqtri 2761 . 2 dom 𝐾 = {π‘₯ ∈ 𝐿 ∣ (π‘₯𝑃( βŠ₯ β€˜π‘₯)) ∈ (𝑉 ↑m 𝑉)}
288, 27elrab2 3687 1 (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( βŠ₯ β€˜π‘‡)):π‘‰βŸΆπ‘‰))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ∩ cin 3948   ↦ cmpt 5232   Γ— cxp 5675  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  Basecbs 17144  proj1cpj1 19503  LSubSpclss 20542  ocvcocv 21213  projcpj 21255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-pj 21258
This theorem is referenced by:  pjfval2  21264  pjdm2  21266  pjf  21268
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