HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hmopidmchi Structured version   Unicode version

Theorem hmopidmchi 26893
Description: An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
hmopidmch.1  |-  T  e. 
HrmOp
hmopidmch.2  |-  ( T  o.  T )  =  T
Assertion
Ref Expression
hmopidmchi  |-  ran  T  e.  CH

Proof of Theorem hmopidmchi
Dummy variables  f 
k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmopidmch.1 . . . 4  |-  T  e. 
HrmOp
2 hmoplin 26684 . . . 4  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 5 . . 3  |-  T  e. 
LinOp
43rnelshi 26801 . 2  |-  ran  T  e.  SH
5 eqid 2467 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
65hilxmet 25935 . . . . . . 7  |-  ( normh  o. 
-h  )  e.  ( *Met `  ~H )
7 eqid 2467 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
87methaus 20891 . . . . . . 7  |-  ( (
normh  o.  -h  )  e.  ( *Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  )
)  e.  Haus )
96, 8mp1i 12 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  Haus )
10 eqid 2467 . . . . . . . . . . . 12  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
1110, 5hhims 25912 . . . . . . . . . . . 12  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
1210, 11, 7hhlm 25939 . . . . . . . . . . 11  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
13 resss 5303 . . . . . . . . . . 11  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1412, 13eqsstri 3539 . . . . . . . . . 10  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1514ssbri 4495 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  f ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) ) x )
1615adantl 466 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) x )
177mopntopon 20810 . . . . . . . . . 10  |-  ( (
normh  o.  -h  )  e.  ( *Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  )
)  e.  (TopOn `  ~H ) )
186, 17mp1i 12 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  (TopOn `  ~H ) )
193lnopfi 26711 . . . . . . . . . . . 12  |-  T : ~H
--> ~H
2019a1i 11 . . . . . . . . . . 11  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T : ~H --> ~H )
2120feqmptd 5927 . . . . . . . . . 10  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T  =  ( y  e.  ~H  |->  ( T `  y ) ) )
22 hmopbdoptHIL 26730 . . . . . . . . . . . . 13  |-  ( T  e.  HrmOp  ->  T  e.  BndLinOp )
231, 22ax-mp 5 . . . . . . . . . . . 12  |-  T  e.  BndLinOp
24 lnopcnbd 26778 . . . . . . . . . . . . 13  |-  ( T  e.  LinOp  ->  ( T  e.  ConOp 
<->  T  e.  BndLinOp ) )
253, 24ax-mp 5 . . . . . . . . . . . 12  |-  ( T  e.  ConOp 
<->  T  e.  BndLinOp )
2623, 25mpbir 209 . . . . . . . . . . 11  |-  T  e. 
ConOp
275, 7hhcno 26646 . . . . . . . . . . 11  |-  ConOp  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) )
2826, 27eleqtri 2553 . . . . . . . . . 10  |-  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) )
2921, 28syl6eqelr 2564 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  ( T `  y ) )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3018cnmptid 20030 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  y )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3110hhnv 25905 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3210hhvs 25910 . . . . . . . . . . 11  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
3311, 7, 32vmcn 25432 . . . . . . . . . 10  |-  ( <. <.  +h  ,  .h  >. , 
normh >.  e.  NrmCVec  ->  -h  e.  ( ( ( MetOpen `  ( normh  o.  -h  )
)  tX  ( MetOpen `  ( normh  o.  -h  )
) )  Cn  ( MetOpen
`  ( normh  o.  -h  ) ) ) )
3431, 33mp1i 12 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  -h  e.  (
( ( MetOpen `  ( normh  o.  -h  ) ) 
tX  ( MetOpen `  ( normh  o.  -h  ) ) )  Cn  ( MetOpen `  ( normh  o.  -h  )
) ) )
3518, 29, 30, 34cnmpt12f 20035 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3616, 35lmcn 19674 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f ) ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) ) ( ( y  e.  ~H  |->  ( ( T `  y
)  -h  y ) ) `  x ) )
37 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f : NN --> ran  T )
384shssii 25953 . . . . . . . . . . . . . 14  |-  ran  T  C_ 
~H
39 fss 5745 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  ran  T  C_  ~H )  ->  f : NN --> ~H )
4037, 38, 39sylancl 662 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f : NN --> ~H )
4140ffvelrnda 6032 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ~H )
42 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  ( T `  y )  =  ( T `  ( f `  k
) ) )
43 id 22 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  y  =  ( f `  k ) )
4442, 43oveq12d 6313 . . . . . . . . . . . . 13  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 ( f `  k ) )  -h  ( f `  k
) ) )
45 eqid 2467 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  =  ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )
46 ovex 6320 . . . . . . . . . . . . 13  |-  ( ( T `  ( f `
 k ) )  -h  ( f `  k ) )  e. 
_V
4744, 45, 46fvmpt 5957 . . . . . . . . . . . 12  |-  ( ( f `  k )  e.  ~H  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  (
f `  k )
)  =  ( ( T `  ( f `
 k ) )  -h  ( f `  k ) ) )
4841, 47syl 16 . . . . . . . . . . 11  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  (
f `  k )
)  =  ( ( T `  ( f `
 k ) )  -h  ( f `  k ) ) )
49 ffn 5737 . . . . . . . . . . . . . . . 16  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
5019, 49ax-mp 5 . . . . . . . . . . . . . . 15  |-  T  Fn  ~H
51 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  ( T `  y )  =  ( T `  ( T `  x ) ) )
52 id 22 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  y  =  ( T `  x ) )
5351, 52eqeq12d 2489 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( T `  x )  ->  (
( T `  y
)  =  y  <->  ( T `  ( T `  x
) )  =  ( T `  x ) ) )
5453ralrn 6035 . . . . . . . . . . . . . . 15  |-  ( T  Fn  ~H  ->  ( A. y  e.  ran  T ( T `  y
)  =  y  <->  A. x  e.  ~H  ( T `  ( T `  x ) )  =  ( T `
 x ) ) )
5550, 54ax-mp 5 . . . . . . . . . . . . . 14  |-  ( A. y  e.  ran  T ( T `  y )  =  y  <->  A. x  e.  ~H  ( T `  ( T `  x ) )  =  ( T `
 x ) )
56 hmopidmch.2 . . . . . . . . . . . . . . . 16  |-  ( T  o.  T )  =  T
5756fveq1i 5873 . . . . . . . . . . . . . . 15  |-  ( ( T  o.  T ) `
 x )  =  ( T `  x
)
5819, 19hocoi 26506 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( T  o.  T
) `  x )  =  ( T `  ( T `  x ) ) )
5957, 58syl5reqr 2523 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( T `  ( T `  x ) )  =  ( T `  x
) )
6055, 59mprgbir 2831 . . . . . . . . . . . . 13  |-  A. y  e.  ran  T ( T `
 y )  =  y
61 ffvelrn 6030 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  k  e.  NN )  ->  ( f `  k )  e.  ran  T )
6261adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ran  T )
6342, 43eqeq12d 2489 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  =  y  <->  ( T `  ( f `  k
) )  =  ( f `  k ) ) )
6463rspccv 3216 . . . . . . . . . . . . 13  |-  ( A. y  e.  ran  T ( T `  y )  =  y  ->  (
( f `  k
)  e.  ran  T  ->  ( T `  (
f `  k )
)  =  ( f `
 k ) ) )
6560, 62, 64mpsyl 63 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  ( T `  ( f `  k ) )  =  ( f `  k
) )
6665, 41eqeltrd 2555 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  ( T `  ( f `  k ) )  e. 
~H )
67 hvsubeq0 25808 . . . . . . . . . . . . 13  |-  ( ( ( T `  (
f `  k )
)  e.  ~H  /\  ( f `  k
)  e.  ~H )  ->  ( ( ( T `
 ( f `  k ) )  -h  ( f `  k
) )  =  0h  <->  ( T `  ( f `
 k ) )  =  ( f `  k ) ) )
6866, 41, 67syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( ( T `  ( f `  k
) )  -h  (
f `  k )
)  =  0h  <->  ( T `  ( f `  k
) )  =  ( f `  k ) ) )
6965, 68mpbird 232 . . . . . . . . . . 11  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( T `  (
f `  k )
)  -h  ( f `
 k ) )  =  0h )
7048, 69eqtrd 2508 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  (
f `  k )
)  =  0h )
71 fvco3 5951 . . . . . . . . . . 11  |-  ( ( f : NN --> ran  T  /\  k  e.  NN )  ->  ( ( ( y  e.  ~H  |->  ( ( T `  y
)  -h  y ) )  o.  f ) `
 k )  =  ( ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) ) `  ( f `  k
) ) )
7271adantlr 714 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )  o.  f ) `  k
)  =  ( ( y  e.  ~H  |->  ( ( T `  y
)  -h  y ) ) `  ( f `
 k ) ) )
73 ax-hv0cl 25743 . . . . . . . . . . . . 13  |-  0h  e.  ~H
7473elexi 3128 . . . . . . . . . . . 12  |-  0h  e.  _V
7574fvconst2 6127 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
( NN  X.  { 0h } ) `  k
)  =  0h )
7675adantl 466 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( NN  X.  { 0h } ) `  k
)  =  0h )
7770, 72, 763eqtr4d 2518 . . . . . . . . 9  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )  o.  f ) `  k
)  =  ( ( NN  X.  { 0h } ) `  k
) )
7877ralrimiva 2881 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  A. k  e.  NN  ( ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f ) `  k )  =  ( ( NN  X.  { 0h } ) `  k
) )
79 ovex 6320 . . . . . . . . . . 11  |-  ( ( T `  y )  -h  y )  e. 
_V
8079, 45fnmpti 5715 . . . . . . . . . 10  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  Fn  ~H
81 fnfco 5756 . . . . . . . . . 10  |-  ( ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y
) )  Fn  ~H  /\  f : NN --> ~H )  ->  ( ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )  o.  f )  Fn  NN )
8280, 40, 81sylancr 663 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f )  Fn  NN )
8374fconst 5777 . . . . . . . . . 10  |-  ( NN 
X.  { 0h }
) : NN --> { 0h }
84 ffn 5737 . . . . . . . . . 10  |-  ( ( NN  X.  { 0h } ) : NN --> { 0h }  ->  ( NN  X.  { 0h }
)  Fn  NN )
8583, 84ax-mp 5 . . . . . . . . 9  |-  ( NN 
X.  { 0h }
)  Fn  NN
86 eqfnfv 5982 . . . . . . . . 9  |-  ( ( ( ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )  o.  f )  Fn  NN  /\  ( NN  X.  { 0h } )  Fn  NN )  ->  ( ( ( y  e.  ~H  |->  ( ( T `  y
)  -h  y ) )  o.  f )  =  ( NN  X.  { 0h } )  <->  A. k  e.  NN  ( ( ( y  e.  ~H  |->  ( ( T `  y
)  -h  y ) )  o.  f ) `
 k )  =  ( ( NN  X.  { 0h } ) `  k ) ) )
8782, 85, 86sylancl 662 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( ( y  e.  ~H  |->  ( ( T `  y
)  -h  y ) )  o.  f )  =  ( NN  X.  { 0h } )  <->  A. k  e.  NN  ( ( ( y  e.  ~H  |->  ( ( T `  y
)  -h  y ) )  o.  f ) `
 k )  =  ( ( NN  X.  { 0h } ) `  k ) ) )
8878, 87mpbird 232 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f )  =  ( NN  X.  { 0h } ) )
89 vex 3121 . . . . . . . . . 10  |-  x  e. 
_V
9089hlimveci 25930 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  x  e.  ~H )
9190adantl 466 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ~H )
92 fveq2 5872 . . . . . . . . . 10  |-  ( y  =  x  ->  ( T `  y )  =  ( T `  x ) )
93 id 22 . . . . . . . . . 10  |-  ( y  =  x  ->  y  =  x )
9492, 93oveq12d 6313 . . . . . . . . 9  |-  ( y  =  x  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 x )  -h  x ) )
95 ovex 6320 . . . . . . . . 9  |-  ( ( T `  x )  -h  x )  e. 
_V
9694, 45, 95fvmpt 5957 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  x
)  =  ( ( T `  x )  -h  x ) )
9791, 96syl 16 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) ) `
 x )  =  ( ( T `  x )  -h  x
) )
9836, 88, 973brtr3d 4482 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) ) ( ( T `  x )  -h  x ) )
9973a1i 11 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  0h  e.  ~H )
100 1zzd 10907 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  1  e.  ZZ )
101 nnuz 11129 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
102101lmconst 19630 . . . . . . 7  |-  ( ( ( MetOpen `  ( normh  o. 
-h  ) )  e.  (TopOn `  ~H )  /\  0h  e.  ~H  /\  1  e.  ZZ )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) 0h )
10318, 99, 100, 102syl3anc 1228 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) ) 0h )
1049, 98, 103lmmo 19749 . . . . 5  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( T `
 x )  -h  x )  =  0h )
10519ffvelrni 6031 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
10691, 105syl 16 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  e.  ~H )
107 hvsubeq0 25808 . . . . . 6  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( ( T `
 x )  -h  x )  =  0h  <->  ( T `  x )  =  x ) )
108106, 91, 107syl2anc 661 . . . . 5  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( ( T `  x )  -h  x )  =  0h  <->  ( T `  x )  =  x ) )
109104, 108mpbid 210 . . . 4  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  =  x )
110 fnfvelrn 6029 . . . . 5  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ran  T
)
11150, 91, 110sylancr 663 . . . 4  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  e.  ran  T )
112109, 111eqeltrrd 2556 . . 3  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
113112gen2 1602 . 2  |-  A. f A. x ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
114 isch2 25964 . 2  |-  ( ran 
T  e.  CH  <->  ( ran  T  e.  SH  /\  A. f A. x ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T ) ) )
1154, 113, 114mpbir2an 918 1  |-  ran  T  e.  CH
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   {csn 4033   <.cop 4039   class class class wbr 4453    |-> cmpt 4511    X. cxp 5003   ran crn 5006    |` cres 5007    o. ccom 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   1c1 9505   NNcn 10548   ZZcz 10876   *Metcxmt 18273   MetOpencmopn 18278  TopOnctopon 19264    Cn ccn 19593   ~~> tclm 19595   Hauscha 19677    tX ctx 19929   NrmCVeccnv 25300   ~Hchil 25659    +h cva 25660    .h csm 25661   normhcno 25663   0hc0v 25664    -h cmv 25665    ~~>v chli 25667   SHcsh 25668   CHcch 25669   ConOpccop 25686   LinOpclo 25687   BndLinOpcbo 25688   HrmOpcho 25690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cc 8827  ax-dc 8838  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584  ax-hilex 25739  ax-hfvadd 25740  ax-hvcom 25741  ax-hvass 25742  ax-hv0cl 25743  ax-hvaddid 25744  ax-hfvmul 25745  ax-hvmulid 25746  ax-hvmulass 25747  ax-hvdistr1 25748  ax-hvdistr2 25749  ax-hvmul0 25750  ax-hfi 25819  ax-his1 25822  ax-his2 25823  ax-his3 25824  ax-his4 25825  ax-hcompl 25942
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-cn 19596  df-cnp 19597  df-lm 19598  df-t1 19683  df-haus 19684  df-cmp 19755  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-fcls 20310  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-cfil 21562  df-cau 21563  df-cmet 21564  df-grpo 25016  df-gid 25017  df-ginv 25018  df-gdiv 25019  df-ablo 25107  df-subgo 25127  df-vc 25262  df-nv 25308  df-va 25311  df-ba 25312  df-sm 25313  df-0v 25314  df-vs 25315  df-nmcv 25316  df-ims 25317  df-dip 25434  df-ssp 25458  df-lno 25482  df-nmoo 25483  df-blo 25484  df-0o 25485  df-ph 25551  df-cbn 25602  df-hlo 25625  df-hnorm 25708  df-hba 25709  df-hvsub 25711  df-hlim 25712  df-hcau 25713  df-sh 25947  df-ch 25962  df-oc 25993  df-ch0 25994  df-shs 26049  df-pjh 26136  df-h0op 26490  df-nmop 26581  df-cnop 26582  df-lnop 26583  df-bdop 26584  df-unop 26585  df-hmop 26586
This theorem is referenced by:  hmopidmpji  26894  hmopidmch  26895
  Copyright terms: Public domain W3C validator