Theorem pjdm 21616

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 6858	. . . . 5 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇))
3	1, 2	oveq12d 7405	. . . 4 ⊢ (𝑥 = 𝑇 → (𝑥𝑃( ⊥ ‘𝑥)) = (𝑇𝑃( ⊥ ‘𝑇)))
4	3	eleq1d 2813	. . 3 ⊢ (𝑥 = 𝑇 → ((𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)) ∈ (𝑉 ↑m 𝑉)))
5		pjfval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
6	5	fvexi 6872	. . . 4 ⊢ 𝑉 ∈ V
7	6, 6	elmap 8844	. . 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 6117	. . . . . . 7 ⊢ ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑m 𝑉)))
10		cnvxp 6130	. . . . . . . 8 ⊢ ◡(V × (𝑉 ↑m 𝑉)) = ((𝑉 ↑m 𝑉) × V)
11	10	ineq2i 4180	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
12	9, 11	eqtri 2752	. . . . . 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 21615	. . . . . . 7 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
18	17	cnveqi 5838	. . . . . 6 ⊢ ◡𝐾 = ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
19		df-res 5650	. . . . . 6 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉)) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
20	12, 18, 19	3eqtr4i 2762	. . . . 5 ⊢ ◡𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
21	20	rneqi 5901	. . . 4 ⊢ ran ◡𝐾 = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
22		dfdm4 5859	. . . 4 ⊢ dom 𝐾 = ran ◡𝐾
23		df-ima 5651	. . . 4 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉)) = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
24	21, 22, 23	3eqtr4i 2762	. . 3 ⊢ dom 𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉))
25		eqid 2729	. . . 4 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
26	25	mptpreima 6211	. . 3 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉)) = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉)}
27	24, 26	eqtri 2752	. 2 ⊢ dom 𝐾 = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉)}
28	8, 27	elrab2 3662	1 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ∩ cin 3913   ↦ cmpt 5188   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  Basecbs 17179  proj₁cpj1 19565  LSubSpclss 20837  ocvcocv 21569  projcpj 21609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-pj 21612

This theorem is referenced by:  pjfval2  21618  pjdm2  21620  pjf  21622