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Theorem pjdm 19212
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 5825 . . . . 5  |-  ( x  =  T  ->  (  ._|_  `  x )  =  (  ._|_  `  T ) )
31, 2oveq12d 6267 . . . 4  |-  ( x  =  T  ->  (
x P (  ._|_  `  x ) )  =  ( T P ( 
._|_  `  T ) ) )
43eleq1d 2490 . . 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 5835 . . . . 5  |-  ( Base `  W )  e.  _V
75, 6eqeltri 2502 . . . 4  |-  V  e. 
_V
87, 7elmap 7455 . . 3  |-  ( ( T P (  ._|_  `  T ) )  e.  ( V  ^m  V
)  <->  ( T P (  ._|_  `  T ) ) : V --> V )
94, 8syl6bb 264 . 2  |-  ( x  =  T  ->  (
( x P ( 
._|_  `  x ) )  e.  ( V  ^m  V )  <->  ( T P (  ._|_  `  T
) ) : V --> V ) )
10 cnvin 5205 . . . . . . 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 5216 . . . . . . . 8  |-  `' ( _V  X.  ( V  ^m  V ) )  =  ( ( V  ^m  V )  X. 
_V )
1211ineq2i 3604 . . . . . . 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 2450 . . . . . 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 19211 . . . . . . 7  |-  K  =  ( ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
1918cnveqi 4971 . . . . . 6  |-  `' K  =  `' ( ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  ( V  ^m  V ) ) )
20 df-res 4808 . . . . . 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 2460 . . . . 5  |-  `' K  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )  |`  ( V  ^m  V
) )
2221rneqi 5023 . . . 4  |-  ran  `' K  =  ran  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )
23 dfdm4 4989 . . . 4  |-  dom  K  =  ran  `' K
24 df-ima 4809 . . . 4  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) ) " ( V  ^m  V ) )  =  ran  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )  |`  ( V  ^m  V ) )
2522, 23, 243eqtr4i 2460 . . 3  |-  dom  K  =  ( `' ( x  e.  L  |->  ( x P (  ._|_  `  x ) ) )
" ( V  ^m  V ) )
26 eqid 2428 . . . 4  |-  ( x  e.  L  |->  ( x P (  ._|_  `  x
) ) )  =  ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) )
2726mptpreima 5290 . . 3  |-  ( `' ( x  e.  L  |->  ( x P ( 
._|_  `  x ) ) ) " ( V  ^m  V ) )  =  { x  e.  L  |  ( x P (  ._|_  `  x
) )  e.  ( V  ^m  V ) }
2825, 27eqtri 2450 . 2  |-  dom  K  =  { x  e.  L  |  ( x P (  ._|_  `  x ) )  e.  ( V  ^m  V ) }
299, 28elrab2 3173 1  |-  ( T  e.  dom  K  <->  ( T  e.  L  /\  ( T P (  ._|_  `  T
) ) : V --> V ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {crab 2718   _Vcvv 3022    i^i cin 3378    |-> cmpt 4425    X. cxp 4794   `'ccnv 4795   dom cdm 4796   ran crn 4797    |` cres 4798   "cima 4799   -->wf 5540   ` cfv 5544  (class class class)co 6249    ^m cmap 7427   Basecbs 15064   proj1cpj1 17230   LSubSpclss 18098   ocvcocv 19165   projcpj 19205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-map 7429  df-pj 19208
This theorem is referenced by:  pjfval2  19214  pjdm2  19216  pjf  19218
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