Theorem pjdm 21745

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 = (proj1‘𝑊)
pjfval.k	⊢ 𝐾 = (proj‘𝑊)

Assertion

Ref	Expression
pjdm	⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))

Proof of Theorem pjdm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5 ⊢ (𝑥 = 𝑇 → 𝑥 = 𝑇)
2		fveq2 6907	. . . . 5 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇))
3	1, 2	oveq12d 7449	. . . 4 ⊢ (𝑥 = 𝑇 → (𝑥𝑃( ⊥ ‘𝑥)) = (𝑇𝑃( ⊥ ‘𝑇)))
4	3	eleq1d 2824	. . 3 ⊢ (𝑥 = 𝑇 → ((𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)) ∈ (𝑉 ↑m 𝑉)))
5		pjfval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
6	5	fvexi 6921	. . . 4 ⊢ 𝑉 ∈ V
7	6, 6	elmap 8910	. . 3 ⊢ ((𝑇𝑃( ⊥ ‘𝑇)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉)
8	4, 7	bitrdi 287	. 2 ⊢ (𝑥 = 𝑇 → ((𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))
9		cnvin 6167	. . . . . . 7 ⊢ ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑m 𝑉)))
10		cnvxp 6179	. . . . . . . 8 ⊢ ◡(V × (𝑉 ↑m 𝑉)) = ((𝑉 ↑m 𝑉) × V)
11	10	ineq2i 4225	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
12	9, 11	eqtri 2763	. . . . . 6 ⊢ ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
13		pjfval.l	. . . . . . . 8 ⊢ 𝐿 = (LSubSp‘𝑊)
14		pjfval.o	. . . . . . . 8 ⊢ ⊥ = (ocv‘𝑊)
15		pjfval.p	. . . . . . . 8 ⊢ 𝑃 = (proj1‘𝑊)
16		pjfval.k	. . . . . . . 8 ⊢ 𝐾 = (proj‘𝑊)
17	5, 13, 14, 15, 16	pjfval 21744	. . . . . . 7 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
18	17	cnveqi 5888	. . . . . 6 ⊢ ◡𝐾 = ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
19		df-res 5701	. . . . . 6 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉)) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
20	12, 18, 19	3eqtr4i 2773	. . . . 5 ⊢ ◡𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
21	20	rneqi 5951	. . . 4 ⊢ ran ◡𝐾 = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
22		dfdm4 5909	. . . 4 ⊢ dom 𝐾 = ran ◡𝐾
23		df-ima 5702	. . . 4 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉)) = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
24	21, 22, 23	3eqtr4i 2773	. . 3 ⊢ dom 𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉))
25		eqid 2735	. . . 4 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
26	25	mptpreima 6260	. . 3 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉)) = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉)}
27	24, 26	eqtri 2763	. 2 ⊢ dom 𝐾 = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉)}
28	8, 27	elrab2 3698	1 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478   ∩ cin 3962   ↦ cmpt 5231   × cxp 5687  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   ↾ cres 5691   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  Basecbs 17245  proj₁cpj1 19668  LSubSpclss 20947  ocvcocv 21696  projcpj 21738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-pj 21741

This theorem is referenced by:  pjfval2  21747  pjdm2  21749  pjf  21751