Theorem pjdm 21750

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 6920	. . . . 5 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇))
3	1, 2	oveq12d 7466	. . . 4 ⊢ (𝑥 = 𝑇 → (𝑥𝑃( ⊥ ‘𝑥)) = (𝑇𝑃( ⊥ ‘𝑇)))
4	3	eleq1d 2829	. . 3 ⊢ (𝑥 = 𝑇 → ((𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)) ∈ (𝑉 ↑m 𝑉)))
5		pjfval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
6	5	fvexi 6934	. . . 4 ⊢ 𝑉 ∈ V
7	6, 6	elmap 8929	. . 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 6176	. . . . . . 7 ⊢ ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑m 𝑉)))
10		cnvxp 6188	. . . . . . . 8 ⊢ ◡(V × (𝑉 ↑m 𝑉)) = ((𝑉 ↑m 𝑉) × V)
11	10	ineq2i 4238	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
12	9, 11	eqtri 2768	. . . . . 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 21749	. . . . . . 7 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
18	17	cnveqi 5899	. . . . . 6 ⊢ ◡𝐾 = ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
19		df-res 5712	. . . . . 6 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉)) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
20	12, 18, 19	3eqtr4i 2778	. . . . 5 ⊢ ◡𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
21	20	rneqi 5962	. . . 4 ⊢ ran ◡𝐾 = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
22		dfdm4 5920	. . . 4 ⊢ dom 𝐾 = ran ◡𝐾
23		df-ima 5713	. . . 4 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉)) = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
24	21, 22, 23	3eqtr4i 2778	. . 3 ⊢ dom 𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉))
25		eqid 2740	. . . 4 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
26	25	mptpreima 6269	. . 3 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉)) = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉)}
27	24, 26	eqtri 2768	. 2 ⊢ dom 𝐾 = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉)}
28	8, 27	elrab2 3711	1 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488   ∩ cin 3975   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Basecbs 17258  proj₁cpj1 19677  LSubSpclss 20952  ocvcocv 21701  projcpj 21743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-pj 21746

This theorem is referenced by:  pjfval2  21752  pjdm2  21754  pjf  21756