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Theorem pjdm 18132
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 5691 . . . . 5  |-  ( x  =  T  ->  (  ._|_  `  x )  =  (  ._|_  `  T ) )
31, 2oveq12d 6109 . . . 4  |-  ( x  =  T  ->  (
x P (  ._|_  `  x ) )  =  ( T P ( 
._|_  `  T ) ) )
43eleq1d 2509 . . 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 5701 . . . . 5  |-  ( Base `  W )  e.  _V
75, 6eqeltri 2513 . . . 4  |-  V  e. 
_V
87, 7elmap 7241 . . 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 5244 . . . . . . 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 5255 . . . . . . . 8  |-  `' ( _V  X.  ( V  ^m  V ) )  =  ( ( V  ^m  V )  X. 
_V )
1211ineq2i 3549 . . . . . . 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 2463 . . . . . 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 18131 . . . . . . 7  |-  K  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
1918cnveqi 5014 . . . . . 6  |-  `' K  =  `' ( ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 df-res 4852 . . . . . 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 2473 . . . . 5  |-  `' K  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  |`  ( V  ^m  V
) )
2221rneqi 5066 . . . 4  |-  ran  `' K  =  ran  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )
23 dfdm4 5032 . . . 4  |-  dom  K  =  ran  `' K
24 df-ima 4853 . . . 4  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) ) " ( V  ^m  V ) )  =  ran  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )
2522, 23, 243eqtr4i 2473 . . 3  |-  dom  K  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
" ( V  ^m  V ) )
26 eqid 2443 . . . 4  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
2726mptpreima 5331 . . 3  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) ) " ( V  ^m  V ) )  =  { x  e.  L  |  ( x P (  ._|_  `  x
) )  e.  ( V  ^m  V ) }
2825, 27eqtri 2463 . 2  |-  dom  K  =  { x  e.  L  |  ( x P (  ._|_  `  x ) )  e.  ( V  ^m  V ) }
299, 28elrab2 3119 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 1369    e. wcel 1756   {crab 2719   _Vcvv 2972    i^i cin 3327    e. cmpt 4350    X. cxp 4838   `'ccnv 4839   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   Basecbs 14174   proj1cpj1 16134   LSubSpclss 17013   ocvcocv 18085   projcpj 18125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-pj 18128
This theorem is referenced by:  pjfval2  18134  pjdm2  18136  pjf  18138
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