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Theorem pjdm 18865
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.
StepHypRef Expression
1 id 22 . . . . 5  |-  ( x  =  T  ->  x  =  T )
2 fveq2 5872 . . . . 5  |-  ( x  =  T  ->  (  ._|_  `  x )  =  (  ._|_  `  T ) )
31, 2oveq12d 6314 . . . 4  |-  ( x  =  T  ->  (
x P (  ._|_  `  x ) )  =  ( T P ( 
._|_  `  T ) ) )
43eleq1d 2526 . . 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 5882 . . . . 5  |-  ( Base `  W )  e.  _V
75, 6eqeltri 2541 . . . 4  |-  V  e. 
_V
87, 7elmap 7466 . . 3  |-  ( ( T P (  ._|_  `  T ) )  e.  ( V  ^m  V
)  <->  ( T P (  ._|_  `  T ) ) : V --> V )
94, 8syl6bb 261 . 2  |-  ( x  =  T  ->  (
( x P ( 
._|_  `  x ) )  e.  ( V  ^m  V )  <->  ( T P (  ._|_  `  T
) ) : V --> V ) )
10 cnvin 5420 . . . . . . 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 5431 . . . . . . . 8  |-  `' ( _V  X.  ( V  ^m  V ) )  =  ( ( V  ^m  V )  X. 
_V )
1211ineq2i 3693 . . . . . . 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 2486 . . . . . 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
)
185, 14, 15, 16, 17pjfval 18864 . . . . . . 7  |-  K  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
1918cnveqi 5187 . . . . . 6  |-  `' K  =  `' ( ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 df-res 5020 . . . . . 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 2496 . . . . 5  |-  `' K  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  |`  ( V  ^m  V
) )
2221rneqi 5239 . . . 4  |-  ran  `' K  =  ran  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )
23 dfdm4 5205 . . . 4  |-  dom  K  =  ran  `' K
24 df-ima 5021 . . . 4  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) ) " ( V  ^m  V ) )  =  ran  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )
2522, 23, 243eqtr4i 2496 . . 3  |-  dom  K  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
" ( V  ^m  V ) )
26 eqid 2457 . . . 4  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
2726mptpreima 5506 . . 3  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) ) " ( V  ^m  V ) )  =  { x  e.  L  |  ( x P (  ._|_  `  x
) )  e.  ( V  ^m  V ) }
2825, 27eqtri 2486 . 2  |-  dom  K  =  { x  e.  L  |  ( x P (  ._|_  `  x ) )  e.  ( V  ^m  V ) }
299, 28elrab2 3259 1  |-  ( T  e.  dom  K  <->  ( T  e.  L  /\  ( T P (  ._|_  `  T
) ) : V --> V ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109    i^i cin 3470    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   Basecbs 14644   proj1cpj1 16782   LSubSpclss 17705   ocvcocv 18818   projcpj 18858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-pj 18861
This theorem is referenced by:  pjfval2  18867  pjdm2  18869  pjf  18871
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