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Theorem hmopidmchi 23607
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 23398 . . . 4  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 8 . . 3  |-  T  e. 
LinOp
43rnelshi 23515 . 2  |-  ran  T  e.  SH
5 eqid 2404 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
65hilxmet 22650 . . . . . . 7  |-  ( normh  o. 
-h  )  e.  ( * Met `  ~H )
7 eqid 2404 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
87methaus 18503 . . . . . . 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 2404 . . . . . . . . . . . 12  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
1110, 5hhims 22627 . . . . . . . . . . . 12  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
1210, 11, 7hhlm 22654 . . . . . . . . . . 11  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
13 resss 5129 . . . . . . . . . . 11  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1412, 13eqsstri 3338 . . . . . . . . . 10  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1514ssbri 4214 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  f ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) ) x )
1615adantl 453 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) x )
177mopntopon 18422 . . . . . . . . . 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 23425 . . . . . . . . . . . 12  |-  T : ~H
--> ~H
2019a1i 11 . . . . . . . . . . 11  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T : ~H --> ~H )
2120feqmptd 5738 . . . . . . . . . 10  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T  =  ( y  e.  ~H  |->  ( T `  y ) ) )
22 hmopbdoptHIL 23444 . . . . . . . . . . . . 13  |-  ( T  e.  HrmOp  ->  T  e.  BndLinOp )
231, 22ax-mp 8 . . . . . . . . . . . 12  |-  T  e.  BndLinOp
24 lnopcnbd 23492 . . . . . . . . . . . . 13  |-  ( T  e.  LinOp  ->  ( T  e.  ConOp 
<->  T  e.  BndLinOp ) )
253, 24ax-mp 8 . . . . . . . . . . . 12  |-  ( T  e.  ConOp 
<->  T  e.  BndLinOp )
2623, 25mpbir 201 . . . . . . . . . . 11  |-  T  e. 
ConOp
275, 7hhcno 23360 . . . . . . . . . . 11  |-  ConOp  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) )
2826, 27eleqtri 2476 . . . . . . . . . 10  |-  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) )
2921, 28syl6eqelr 2493 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  ( T `  y ) )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3018cnmptid 17646 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  y )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3110hhnv 22620 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3210hhvs 22625 . . . . . . . . . . 11  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
3311, 7, 32vmcn 22148 . . . . . . . . . 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 17651 . . . . . . . 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 17323 . . . . . . 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 444 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f : NN --> ran  T )
384shssii 22668 . . . . . . . . . . . . . 14  |-  ran  T  C_ 
~H
39 fss 5558 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  ran  T  C_  ~H )  ->  f : NN --> ~H )
4037, 38, 39sylancl 644 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f : NN --> ~H )
4140ffvelrnda 5829 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ~H )
42 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  ( T `  y )  =  ( T `  ( f `  k
) ) )
43 id 20 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  y  =  ( f `  k ) )
4442, 43oveq12d 6058 . . . . . . . . . . . . 13  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 ( f `  k ) )  -h  ( f `  k
) ) )
45 eqid 2404 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  =  ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )
46 ovex 6065 . . . . . . . . . . . . 13  |-  ( ( T `  ( f `
 k ) )  -h  ( f `  k ) )  e. 
_V
4744, 45, 46fvmpt 5765 . . . . . . . . . . . 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 5550 . . . . . . . . . . . . . . . 16  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
5019, 49ax-mp 8 . . . . . . . . . . . . . . 15  |-  T  Fn  ~H
51 fveq2 5687 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  ( T `  y )  =  ( T `  ( T `  x ) ) )
52 id 20 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  y  =  ( T `  x ) )
5351, 52eqeq12d 2418 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( T `  x )  ->  (
( T `  y
)  =  y  <->  ( T `  ( T `  x
) )  =  ( T `  x ) ) )
5453ralrn 5832 . . . . . . . . . . . . . . 15  |-  ( T  Fn  ~H  ->  ( A. y  e.  ran  T ( T `  y
)  =  y  <->  A. x  e.  ~H  ( T `  ( T `  x ) )  =  ( T `
 x ) ) )
5550, 54ax-mp 8 . . . . . . . . . . . . . 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 5688 . . . . . . . . . . . . . . 15  |-  ( ( T  o.  T ) `
 x )  =  ( T `  x
)
5819, 19hocoi 23220 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( T  o.  T
) `  x )  =  ( T `  ( T `  x ) ) )
5957, 58syl5reqr 2451 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( T `  ( T `  x ) )  =  ( T `  x
) )
6055, 59mprgbir 2736 . . . . . . . . . . . . 13  |-  A. y  e.  ran  T ( T `
 y )  =  y
61 ffvelrn 5827 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  k  e.  NN )  ->  ( f `  k )  e.  ran  T )
6261adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ran  T )
6342, 43eqeq12d 2418 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  =  y  <->  ( T `  ( f `  k
) )  =  ( f `  k ) ) )
6463rspccv 3009 . . . . . . . . . . . . 13  |-  ( A. y  e.  ran  T ( T `  y )  =  y  ->  (
( f `  k
)  e.  ran  T  ->  ( T `  (
f `  k )
)  =  ( f `
 k ) ) )
6560, 62, 64mpsyl 61 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  ( T `  ( f `  k ) )  =  ( f `  k
) )
6665, 41eqeltrd 2478 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  ( T `  ( f `  k ) )  e. 
~H )
67 hvsubeq0 22523 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 224 . . . . . . . . . . 11  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( T `  (
f `  k )
)  -h  ( f `
 k ) )  =  0h )
7048, 69eqtrd 2436 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  (
f `  k )
)  =  0h )
71 fvco3 5759 . . . . . . . . . . 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 696 . . . . . . . . . 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 22459 . . . . . . . . . . . . 13  |-  0h  e.  ~H
7473elexi 2925 . . . . . . . . . . . 12  |-  0h  e.  _V
7574fvconst2 5906 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
( NN  X.  { 0h } ) `  k
)  =  0h )
7675adantl 453 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( NN  X.  { 0h } ) `  k
)  =  0h )
7770, 72, 763eqtr4d 2446 . . . . . . . . 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 2749 . . . . . . . 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 6065 . . . . . . . . . . 11  |-  ( ( T `  y )  -h  y )  e. 
_V
8079, 45fnmpti 5532 . . . . . . . . . 10  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  Fn  ~H
81 fnfco 5568 . . . . . . . . . 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 645 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f )  Fn  NN )
8374fconst 5588 . . . . . . . . . 10  |-  ( NN 
X.  { 0h }
) : NN --> { 0h }
84 ffn 5550 . . . . . . . . . 10  |-  ( ( NN  X.  { 0h } ) : NN --> { 0h }  ->  ( NN  X.  { 0h }
)  Fn  NN )
8583, 84ax-mp 8 . . . . . . . . 9  |-  ( NN 
X.  { 0h }
)  Fn  NN
86 eqfnfv 5786 . . . . . . . . 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 644 . . . . . . . 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 224 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f )  =  ( NN  X.  { 0h } ) )
89 vex 2919 . . . . . . . . . 10  |-  x  e. 
_V
9089hlimveci 22645 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  x  e.  ~H )
9190adantl 453 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ~H )
92 fveq2 5687 . . . . . . . . . 10  |-  ( y  =  x  ->  ( T `  y )  =  ( T `  x ) )
93 id 20 . . . . . . . . . 10  |-  ( y  =  x  ->  y  =  x )
9492, 93oveq12d 6058 . . . . . . . . 9  |-  ( y  =  x  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 x )  -h  x ) )
95 ovex 6065 . . . . . . . . 9  |-  ( ( T `  x )  -h  x )  e. 
_V
9694, 45, 95fvmpt 5765 . . . . . . . 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 4201 . . . . . 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 1z 10267 . . . . . . . 8  |-  1  e.  ZZ
101100a1i 11 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  1  e.  ZZ )
102 nnuz 10477 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
103102lmconst 17279 . . . . . . 7  |-  ( ( ( MetOpen `  ( normh  o. 
-h  ) )  e.  (TopOn `  ~H )  /\  0h  e.  ~H  /\  1  e.  ZZ )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) 0h )
10418, 99, 101, 103syl3anc 1184 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) ) 0h )
1059, 98, 104lmmo 17398 . . . . 5  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( T `
 x )  -h  x )  =  0h )
10619ffvelrni 5828 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
10791, 106syl 16 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  e.  ~H )
108 hvsubeq0 22523 . . . . . 6  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( ( T `
 x )  -h  x )  =  0h  <->  ( T `  x )  =  x ) )
109107, 91, 108syl2anc 643 . . . . 5  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( ( T `  x )  -h  x )  =  0h  <->  ( T `  x )  =  x ) )
110105, 109mpbid 202 . . . 4  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  =  x )
111 fnfvelrn 5826 . . . . 5  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ran  T
)
11250, 91, 111sylancr 645 . . . 4  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  e.  ran  T )
113110, 112eqeltrrd 2479 . . 3  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
114113gen2 1553 . 2  |-  A. f A. x ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
115 isch2 22679 . 2  |-  ( ran 
T  e.  CH  <->  ( ran  T  e.  SH  /\  A. f A. x ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T ) ) )
1164, 114, 115mpbir2an 887 1  |-  ran  T  e.  CH
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   {csn 3774   <.cop 3777   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   ran crn 4838    |` cres 4839    o. ccom 4841    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   1c1 8947   NNcn 9956   ZZcz 10238   * Metcxmt 16641   MetOpencmopn 16646  TopOnctopon 16914    Cn ccn 17242   ~~> tclm 17244   Hauscha 17326    tX ctx 17545   NrmCVeccnv 22016   ~Hchil 22375    +h cva 22376    .h csm 22377   normhcno 22379   0hc0v 22380    -h cmv 22381    ~~>v chli 22383   SHcsh 22384   CHcch 22385   ConOpccop 22402   LinOpclo 22403   BndLinOpcbo 22404   HrmOpcho 22406
This theorem is referenced by:  hmopidmpji  23608  hmopidmch  23609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-dc 8282  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540  ax-hcompl 22657
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-cnp 17246  df-lm 17247  df-t1 17332  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-fcls 17926  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-cfil 19161  df-cau 19162  df-cmet 19163  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-subgo 21843  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-dip 22150  df-ssp 22174  df-lno 22198  df-nmoo 22199  df-blo 22200  df-0o 22201  df-ph 22267  df-cbn 22318  df-hlo 22341  df-hnorm 22424  df-hba 22425  df-hvsub 22427  df-hlim 22428  df-hcau 22429  df-sh 22662  df-ch 22677  df-oc 22707  df-ch0 22708  df-shs 22763  df-pjh 22850  df-h0op 23204  df-nmop 23295  df-cnop 23296  df-lnop 23297  df-bdop 23298  df-unop 23299  df-hmop 23300
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