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Theorem hmopidmchi 25734
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 25525 . . . 4  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 5 . . 3  |-  T  e. 
LinOp
43rnelshi 25642 . 2  |-  ran  T  e.  SH
5 eqid 2454 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
65hilxmet 24776 . . . . . . 7  |-  ( normh  o. 
-h  )  e.  ( *Met `  ~H )
7 eqid 2454 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
87methaus 20237 . . . . . . 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 2454 . . . . . . . . . . . 12  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
1110, 5hhims 24753 . . . . . . . . . . . 12  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
1210, 11, 7hhlm 24780 . . . . . . . . . . 11  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
13 resss 5245 . . . . . . . . . . 11  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1412, 13eqsstri 3497 . . . . . . . . . 10  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1514ssbri 4445 . . . . . . . . 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 20156 . . . . . . . . . 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 25552 . . . . . . . . . . . 12  |-  T : ~H
--> ~H
2019a1i 11 . . . . . . . . . . 11  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T : ~H --> ~H )
2120feqmptd 5856 . . . . . . . . . 10  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T  =  ( y  e.  ~H  |->  ( T `  y ) ) )
22 hmopbdoptHIL 25571 . . . . . . . . . . . . 13  |-  ( T  e.  HrmOp  ->  T  e.  BndLinOp )
231, 22ax-mp 5 . . . . . . . . . . . 12  |-  T  e.  BndLinOp
24 lnopcnbd 25619 . . . . . . . . . . . . 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 25487 . . . . . . . . . . 11  |-  ConOp  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) )
2826, 27eleqtri 2540 . . . . . . . . . 10  |-  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) )
2921, 28syl6eqelr 2551 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  ( T `  y ) )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3018cnmptid 19376 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  y )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3110hhnv 24746 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3210hhvs 24751 . . . . . . . . . . 11  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
3311, 7, 32vmcn 24273 . . . . . . . . . 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 19381 . . . . . . . 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 19051 . . . . . . 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 24794 . . . . . . . . . . . . . 14  |-  ran  T  C_ 
~H
39 fss 5678 . . . . . . . . . . . . . 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 5955 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ~H )
42 fveq2 5802 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  ( T `  y )  =  ( T `  ( f `  k
) ) )
43 id 22 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  y  =  ( f `  k ) )
4442, 43oveq12d 6221 . . . . . . . . . . . . 13  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 ( f `  k ) )  -h  ( f `  k
) ) )
45 eqid 2454 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  =  ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )
46 ovex 6228 . . . . . . . . . . . . 13  |-  ( ( T `  ( f `
 k ) )  -h  ( f `  k ) )  e. 
_V
4744, 45, 46fvmpt 5886 . . . . . . . . . . . 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 5670 . . . . . . . . . . . . . . . 16  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
5019, 49ax-mp 5 . . . . . . . . . . . . . . 15  |-  T  Fn  ~H
51 fveq2 5802 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  ( T `  y )  =  ( T `  ( T `  x ) ) )
52 id 22 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  y  =  ( T `  x ) )
5351, 52eqeq12d 2476 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( T `  x )  ->  (
( T `  y
)  =  y  <->  ( T `  ( T `  x
) )  =  ( T `  x ) ) )
5453ralrn 5958 . . . . . . . . . . . . . . 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 5803 . . . . . . . . . . . . . . 15  |-  ( ( T  o.  T ) `
 x )  =  ( T `  x
)
5819, 19hocoi 25347 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( T  o.  T
) `  x )  =  ( T `  ( T `  x ) ) )
5957, 58syl5reqr 2510 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( T `  ( T `  x ) )  =  ( T `  x
) )
6055, 59mprgbir 2904 . . . . . . . . . . . . 13  |-  A. y  e.  ran  T ( T `
 y )  =  y
61 ffvelrn 5953 . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  =  y  <->  ( T `  ( f `  k
) )  =  ( f `  k ) ) )
6463rspccv 3176 . . . . . . . . . . . . 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 2542 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  ( T `  ( f `  k ) )  e. 
~H )
67 hvsubeq0 24649 . . . . . . . . . . . . 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 2495 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  (
f `  k )
)  =  0h )
71 fvco3 5880 . . . . . . . . . . 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 24584 . . . . . . . . . . . . 13  |-  0h  e.  ~H
7473elexi 3088 . . . . . . . . . . . 12  |-  0h  e.  _V
7574fvconst2 6045 . . . . . . . . . . 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 2505 . . . . . . . . 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 2830 . . . . . . . 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 6228 . . . . . . . . . . 11  |-  ( ( T `  y )  -h  y )  e. 
_V
8079, 45fnmpti 5650 . . . . . . . . . 10  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  Fn  ~H
81 fnfco 5688 . . . . . . . . . 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 5707 . . . . . . . . . 10  |-  ( NN 
X.  { 0h }
) : NN --> { 0h }
84 ffn 5670 . . . . . . . . . 10  |-  ( ( NN  X.  { 0h } ) : NN --> { 0h }  ->  ( NN  X.  { 0h }
)  Fn  NN )
8583, 84ax-mp 5 . . . . . . . . 9  |-  ( NN 
X.  { 0h }
)  Fn  NN
86 eqfnfv 5909 . . . . . . . . 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 3081 . . . . . . . . . 10  |-  x  e. 
_V
9089hlimveci 24771 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  x  e.  ~H )
9190adantl 466 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ~H )
92 fveq2 5802 . . . . . . . . . 10  |-  ( y  =  x  ->  ( T `  y )  =  ( T `  x ) )
93 id 22 . . . . . . . . . 10  |-  ( y  =  x  ->  y  =  x )
9492, 93oveq12d 6221 . . . . . . . . 9  |-  ( y  =  x  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 x )  -h  x ) )
95 ovex 6228 . . . . . . . . 9  |-  ( ( T `  x )  -h  x )  e. 
_V
9694, 45, 95fvmpt 5886 . . . . . . . 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 4432 . . . . . 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 10792 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  1  e.  ZZ )
101 nnuz 11011 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
102101lmconst 19007 . . . . . . 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 1219 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) ) 0h )
1049, 98, 103lmmo 19126 . . . . 5  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( T `
 x )  -h  x )  =  0h )
10519ffvelrni 5954 . . . . . . 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 24649 . . . . . 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 5952 . . . . 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 2543 . . 3  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
113112gen2 1593 . 2  |-  A. f A. x ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
114 isch2 24805 . 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 911 1  |-  ran  T  e.  CH
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   A.wral 2799    C_ wss 3439   {csn 3988   <.cop 3994   class class class wbr 4403    |-> cmpt 4461    X. cxp 4949   ran crn 4952    |` cres 4953    o. ccom 4955    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    ^m cmap 7327   1c1 9398   NNcn 10437   ZZcz 10761   *Metcxmt 17936   MetOpencmopn 17941  TopOnctopon 18641    Cn ccn 18970   ~~> tclm 18972   Hauscha 19054    tX ctx 19275   NrmCVeccnv 24141   ~Hchil 24500    +h cva 24501    .h csm 24502   normhcno 24504   0hc0v 24505    -h cmv 24506    ~~>v chli 24508   SHcsh 24509   CHcch 24510   ConOpccop 24527   LinOpclo 24528   BndLinOpcbo 24529   HrmOpcho 24531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cc 8719  ax-dc 8730  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475  ax-addf 9476  ax-mulf 9477  ax-hilex 24580  ax-hfvadd 24581  ax-hvcom 24582  ax-hvass 24583  ax-hv0cl 24584  ax-hvaddid 24585  ax-hfvmul 24586  ax-hvmulid 24587  ax-hvmulass 24588  ax-hvdistr1 24589  ax-hvdistr2 24590  ax-hvmul0 24591  ax-hfi 24660  ax-his1 24663  ax-his2 24664  ax-his3 24665  ax-his4 24666  ax-hcompl 24783
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-er 7214  df-map 7329  df-pm 7330  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7776  df-sup 7806  df-oi 7839  df-card 8224  df-acn 8227  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-q 11069  df-rp 11107  df-xneg 11204  df-xadd 11205  df-xmul 11206  df-ioo 11419  df-ico 11421  df-icc 11422  df-fz 11559  df-fzo 11670  df-fl 11763  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-rlim 13089  df-sum 13286  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-starv 14376  df-sca 14377  df-vsca 14378  df-ip 14379  df-tset 14380  df-ple 14381  df-ds 14383  df-unif 14384  df-hom 14385  df-cco 14386  df-rest 14484  df-topn 14485  df-0g 14503  df-gsum 14504  df-topgen 14505  df-pt 14506  df-prds 14509  df-xrs 14563  df-qtop 14568  df-imas 14569  df-xps 14571  df-mre 14647  df-mrc 14648  df-acs 14650  df-mnd 15538  df-submnd 15588  df-mulg 15671  df-cntz 15958  df-cmn 16404  df-psmet 17944  df-xmet 17945  df-met 17946  df-bl 17947  df-mopn 17948  df-fbas 17949  df-fg 17950  df-cnfld 17954  df-top 18645  df-bases 18647  df-topon 18648  df-topsp 18649  df-cld 18765  df-ntr 18766  df-cls 18767  df-nei 18844  df-cn 18973  df-cnp 18974  df-lm 18975  df-t1 19060  df-haus 19061  df-cmp 19132  df-tx 19277  df-hmeo 19470  df-fil 19561  df-fm 19653  df-flim 19654  df-flf 19655  df-fcls 19656  df-xms 20037  df-ms 20038  df-tms 20039  df-cncf 20596  df-cfil 20908  df-cau 20909  df-cmet 20910  df-grpo 23857  df-gid 23858  df-ginv 23859  df-gdiv 23860  df-ablo 23948  df-subgo 23968  df-vc 24103  df-nv 24149  df-va 24152  df-ba 24153  df-sm 24154  df-0v 24155  df-vs 24156  df-nmcv 24157  df-ims 24158  df-dip 24275  df-ssp 24299  df-lno 24323  df-nmoo 24324  df-blo 24325  df-0o 24326  df-ph 24392  df-cbn 24443  df-hlo 24466  df-hnorm 24549  df-hba 24550  df-hvsub 24552  df-hlim 24553  df-hcau 24554  df-sh 24788  df-ch 24803  df-oc 24834  df-ch0 24835  df-shs 24890  df-pjh 24977  df-h0op 25331  df-nmop 25422  df-cnop 25423  df-lnop 25424  df-bdop 25425  df-unop 25426  df-hmop 25427
This theorem is referenced by:  hmopidmpji  25735  hmopidmch  25736
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