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 = (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 6103	. . . . 5 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇))
3	1, 2	oveq12d 6567	. . . 4 ⊢ (𝑥 = 𝑇 → (𝑥𝑃( ⊥ ‘𝑥)) = (𝑇𝑃( ⊥ ‘𝑇)))
4	3	eleq1d 2672	. . 3 ⊢ (𝑥 = 𝑇 → ((𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑𝑚 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)) ∈ (𝑉 ↑𝑚 𝑉)))
5		pjfval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
6		fvex 6113	. . . . 5 ⊢ (Base‘𝑊) ∈ V
7	5, 6	eqeltri 2684	. . . 4 ⊢ 𝑉 ∈ V
8	7, 7	elmap 7772	. . 3 ⊢ ((𝑇𝑃( ⊥ ‘𝑇)) ∈ (𝑉 ↑𝑚 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉)
9	4, 8	syl6bb 275	. 2 ⊢ (𝑥 = 𝑇 → ((𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑𝑚 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))
10		cnvin 5459	. . . . . . 7 ⊢ ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑𝑚 𝑉)))
11		cnvxp 5470	. . . . . . . 8 ⊢ ◡(V × (𝑉 ↑𝑚 𝑉)) = ((𝑉 ↑𝑚 𝑉) × V)
12	11	ineq2i 3773	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑𝑚 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑𝑚 𝑉) × V))
13	10, 12	eqtri 2632	. . . . . 6 ⊢ ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑𝑚 𝑉) × V))
14		pjfval.l	. . . . . . . 8 ⊢ 𝐿 = (LSubSp‘𝑊)
15		pjfval.o	. . . . . . . 8 ⊢ ⊥ = (ocv‘𝑊)
16		pjfval.p	. . . . . . . 8 ⊢ 𝑃 = (proj1‘𝑊)
17		pjfval.k	. . . . . . . 8 ⊢ 𝐾 = (proj‘𝑊)
18	5, 14, 15, 16, 17	pjfval 19869	. . . . . . 7 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉)))
19	18	cnveqi 5219	. . . . . 6 ⊢ ◡𝐾 = ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑𝑚 𝑉)))
20		df-res 5050	. . . . . 6 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑𝑚 𝑉)) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑𝑚 𝑉) × V))
21	13, 19, 20	3eqtr4i 2642	. . . . 5 ⊢ ◡𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑𝑚 𝑉))
22	21	rneqi 5273	. . . 4 ⊢ ran ◡𝐾 = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑𝑚 𝑉))
23		dfdm4 5238	. . . 4 ⊢ dom 𝐾 = ran ◡𝐾
24		df-ima 5051	. . . 4 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑𝑚 𝑉)) = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑𝑚 𝑉))
25	22, 23, 24	3eqtr4i 2642	. . 3 ⊢ dom 𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑𝑚 𝑉))
26		eqid 2610	. . . 4 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
27	26	mptpreima 5545	. . 3 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑𝑚 𝑉)) = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑𝑚 𝑉)}
28	25, 27	eqtri 2632	. 2 ⊢ dom 𝐾 = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑𝑚 𝑉)}
29	9, 28	elrab2 3333	1 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))

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