Theorem pjdm 22013

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 23	. . . . 5 ⊢ (𝑥 = 𝑇 → 𝑥 = 𝑇)
2		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝑇 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑇))
3	1, 2	oveq12d 7438	. . . 4 ⊢ (𝑥 = 𝑇 → (𝑥𝑃( ⊥ ‘𝑥)) = (𝑇𝑃( ⊥ ‘𝑇)))
4	3	eleq1d 2846	. . 3 ⊢ (𝑥 = 𝑇 → ((𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)) ∈ (𝑉 ↑m 𝑉)))
5		pjfval.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
6	5	fvexi 6899	. . . 4 ⊢ 𝑉 ∈ V
7	6, 6	elmap 8899	. . 3 ⊢ ((𝑇𝑃( ⊥ ‘𝑇)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉)
8	4, 7	bitrdi 290	. 2 ⊢ (𝑥 = 𝑇 → ((𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉) ↔ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))
9		cnvin 6135	. . . . . . 7 ⊢ ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑m 𝑉)))
10		cnvxp 6147	. . . . . . . 8 ⊢ ◡(V × (𝑉 ↑m 𝑉)) = ((𝑉 ↑m 𝑉) × V)
11	10	ineq2i 4163	. . . . . . 7 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ◡(V × (𝑉 ↑m 𝑉))) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
12	9, 11	eqtri 2784	. . . . . 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 22012	. . . . . . 7 ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
18	17	cnveqi 5852	. . . . . 6 ⊢ ◡𝐾 = ◡((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
19		df-res 5663	. . . . . 6 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉)) = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ ((𝑉 ↑m 𝑉) × V))
20	12, 18, 19	3eqtr4i 2794	. . . . 5 ⊢ ◡𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
21	20	rneqi 5919	. . . 4 ⊢ ran ◡𝐾 = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
22		dfdm4 5877	. . . 4 ⊢ dom 𝐾 = ran ◡𝐾
23		df-ima 5664	. . . 4 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉)) = ran (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ↾ (𝑉 ↑m 𝑉))
24	21, 22, 23	3eqtr4i 2794	. . 3 ⊢ dom 𝐾 = (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉))
25		eqid 2761	. . . 4 ⊢ (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) = (𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
26	25	mptpreima 6239	. . 3 ⊢ (◡(𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) “ (𝑉 ↑m 𝑉)) = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉)}
27	24, 26	eqtri 2784	. 2 ⊢ dom 𝐾 = {𝑥 ∈ 𝐿 ∣ (𝑥𝑃( ⊥ ‘𝑥)) ∈ (𝑉 ↑m 𝑉)}
28	8, 27	elrab2 3649	1 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451   ∩ cin 3898   ↦ cmpt 5186   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653   “ cima 5654  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8847  Basecbs 17387  proj₁cpj1 19849  LSubSpclss 21206  ocvcocv 21966  projcpj 22006

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-pj 22009

This theorem is used by:  pjfval2  22015  pjdm2  22017  pjf  22019