Theorem pjdm 20051

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 e. dom K <-> ( T e. L /\ ( T P ( ._|_ ` 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 6191	. . . . 5 |- ( x = T -> ( ._|_ ` x ) = ( ._|_ ` T ) )
3	1, 2	oveq12d 6668	. . . 4 |- ( x = T -> ( x P ( ._|_ ` x ) ) = ( T P ( ._|_ ` T ) ) )
4	3	eleq1d 2686	. . 3 |- ( x = T -> ( ( x P ( ._|_ ` x ) ) e. ( V ^m V ) <-> ( T P ( ._|_ ` T ) ) e. ( V ^m V ) ) )
5		pjfval.v	. . . . 5 |- V = ( Base ` W )
6		fvex 6201	. . . . 5 |- ( Base ` W ) e. _V
7	5, 6	eqeltri 2697	. . . 4 |- V e. _V
8	7, 7	elmap 7886	. . 3 |- ( ( T P ( ._|_ ` T ) ) e. ( V ^m V ) <-> ( T P ( ._|_ ` T ) ) : V --> V )
9	4, 8	syl6bb 276	. 2 |- ( x = T -> ( ( x P ( ._|_ ` x ) ) e. ( V ^m V ) <-> ( T P ( ._|_ ` T ) ) : V --> V ) )
10		cnvin 5540	. . . . . . 7 |- `' ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) = ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i `' ( _V X. ( V ^m V ) ) )
11		cnvxp 5551	. . . . . . . 8 |- `' ( _V X. ( V ^m V ) ) = ( ( V ^m V ) X. _V )
12	11	ineq2i 3811	. . . . . . 7 |- ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i `' ( _V X. ( V ^m V ) ) ) = ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( ( V ^m V ) X. _V ) )
13	10, 12	eqtri 2644	. . . . . 6 |- `' ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) ) = ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( ( V ^m V ) X. _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 20050	. . . . . . 7 |- K = ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) )
19	18	cnveqi 5297	. . . . . 6 |- `' K = `' ( ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( V ^m V ) ) )
20		df-res 5126	. . . . . 6 |- ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) |` ( V ^m V ) ) = ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) i^i ( ( V ^m V ) X. _V ) )
21	13, 19, 20	3eqtr4i 2654	. . . . 5 |- `' K = ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) |` ( V ^m V ) )
22	21	rneqi 5352	. . . 4 |- ran `' K = ran ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) |` ( V ^m V ) )
23		dfdm4 5316	. . . 4 |- dom K = ran `' K
24		df-ima 5127	. . . 4 |- ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) " ( V ^m V ) ) = ran ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) |` ( V ^m V ) )
25	22, 23, 24	3eqtr4i 2654	. . 3 |- dom K = ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) " ( V ^m V ) )
26		eqid 2622	. . . 4 |- ( x e. L |-> ( x P ( ._|_ ` x ) ) ) = ( x e. L |-> ( x P ( ._|_ ` x ) ) )
27	26	mptpreima 5628	. . 3 |- ( `' ( x e. L |-> ( x P ( ._|_ ` x ) ) ) " ( V ^m V ) ) = { x e. L | ( x P ( ._|_ ` x ) ) e. ( V ^m V ) }
28	25, 27	eqtri 2644	. 2 |- dom K = { x e. L | ( x P ( ._|_ ` x ) ) e. ( V ^m V ) }
29	9, 28	elrab2 3366	1 |- ( T e. dom K <-> ( T e. L /\ ( T P ( ._|_ ` T ) ) : V --> V ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   proj1cpj1 18050   LSubSpclss 18932   ocvcocv 20004   projcpj 20044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-pj 20047

This theorem is referenced by:  pjfval2  20053  pjdm2  20055  pjf  20057