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Theorem pjdm 16934
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  =  ( proj 1 `  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.
StepHypRef Expression
1 id 20 . . . . 5  |-  ( x  =  T  ->  x  =  T )
2 fveq2 5728 . . . . 5  |-  ( x  =  T  ->  (  ._|_  `  x )  =  (  ._|_  `  T ) )
31, 2oveq12d 6099 . . . 4  |-  ( x  =  T  ->  (
x P (  ._|_  `  x ) )  =  ( T P ( 
._|_  `  T ) ) )
43eleq1d 2502 . . 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 5742 . . . . 5  |-  ( Base `  W )  e.  _V
75, 6eqeltri 2506 . . . 4  |-  V  e. 
_V
87, 7elmap 7042 . . 3  |-  ( ( T P (  ._|_  `  T ) )  e.  ( V  ^m  V
)  <->  ( T P (  ._|_  `  T ) ) : V --> V )
94, 8syl6bb 253 . 2  |-  ( x  =  T  ->  (
( x P ( 
._|_  `  x ) )  e.  ( V  ^m  V )  <->  ( T P (  ._|_  `  T
) ) : V --> V ) )
10 cnvin 5279 . . . . . . 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 5290 . . . . . . . 8  |-  `' ( _V  X.  ( V  ^m  V ) )  =  ( ( V  ^m  V )  X. 
_V )
1211ineq2i 3539 . . . . . . 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 ) )
1310, 12eqtri 2456 . . . . . 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  =  ( proj 1 `  W )
17 pjfval.k . . . . . . . 8  |-  K  =  ( proj `  W
)
185, 14, 15, 16, 17pjfval 16933 . . . . . . 7  |-  K  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
1918cnveqi 5047 . . . . . 6  |-  `' K  =  `' ( ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 df-res 4890 . . . . . 6  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( ( V  ^m  V )  X.  _V ) )
2113, 19, 203eqtr4i 2466 . . . . 5  |-  `' K  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  |`  ( V  ^m  V
) )
2221rneqi 5096 . . . 4  |-  ran  `' K  =  ran  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )
23 dfdm4 5063 . . . 4  |-  dom  K  =  ran  `' K
24 df-ima 4891 . . . 4  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) ) " ( V  ^m  V ) )  =  ran  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )
2522, 23, 243eqtr4i 2466 . . 3  |-  dom  K  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
" ( V  ^m  V ) )
26 eqid 2436 . . . 4  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
2726mptpreima 5363 . . 3  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) ) " ( V  ^m  V ) )  =  { x  e.  L  |  ( x P (  ._|_  `  x
) )  e.  ( V  ^m  V ) }
2825, 27eqtri 2456 . 2  |-  dom  K  =  { x  e.  L  |  ( x P (  ._|_  `  x ) )  e.  ( V  ^m  V ) }
299, 28elrab2 3094 1  |-  ( T  e.  dom  K  <->  ( T  e.  L  /\  ( T P (  ._|_  `  T
) ) : V --> V ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    i^i cin 3319    e. cmpt 4266    X. cxp 4876   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Basecbs 13469   proj 1cpj1 15269   LSubSpclss 16008   ocvcocv 16887   projcpj 16927
This theorem is referenced by:  pjfval2  16936  pjdm2  16938  pjf  16940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-pj 16930
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