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Theorem hmopidmchi 22727
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 22518 . . . 4  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 8 . . 3  |-  T  e. 
LinOp
43rnelshi 22635 . 2  |-  ran  T  e.  SH
5 eqid 2284 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
65hilxmet 21770 . . . . . . 7  |-  ( normh  o. 
-h  )  e.  ( * Met `  ~H )
7 eqid 2284 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
87methaus 18062 . . . . . . 7  |-  ( (
normh  o.  -h  )  e.  ( * Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  Haus )
96, 8mp1i 11 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  Haus )
10 eqid 2284 . . . . . . . . . . . 12  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
1110, 5hhims 21747 . . . . . . . . . . . 12  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
1210, 11, 7hhlm 21774 . . . . . . . . . . 11  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
13 resss 4978 . . . . . . . . . . 11  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1412, 13eqsstri 3209 . . . . . . . . . 10  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1514ssbri 4066 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  f ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) ) x )
1615adantl 452 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) x )
177mopntopon 17981 . . . . . . . . . 10  |-  ( (
normh  o.  -h  )  e.  ( * Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  (TopOn `  ~H ) )
186, 17mp1i 11 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  (TopOn `  ~H ) )
193lnopfi 22545 . . . . . . . . . . . 12  |-  T : ~H
--> ~H
2019a1i 10 . . . . . . . . . . 11  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T : ~H --> ~H )
2120feqmptd 5537 . . . . . . . . . 10  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T  =  ( y  e.  ~H  |->  ( T `  y ) ) )
22 hmopbdoptHIL 22564 . . . . . . . . . . . . 13  |-  ( T  e.  HrmOp  ->  T  e.  BndLinOp )
231, 22ax-mp 8 . . . . . . . . . . . 12  |-  T  e.  BndLinOp
24 lnopcnbd 22612 . . . . . . . . . . . . 13  |-  ( T  e.  LinOp  ->  ( T  e.  ConOp 
<->  T  e.  BndLinOp ) )
253, 24ax-mp 8 . . . . . . . . . . . 12  |-  ( T  e.  ConOp 
<->  T  e.  BndLinOp )
2623, 25mpbir 200 . . . . . . . . . . 11  |-  T  e. 
ConOp
275, 7hhcno 22480 . . . . . . . . . . 11  |-  ConOp  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) )
2826, 27eleqtri 2356 . . . . . . . . . 10  |-  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) )
2921, 28syl6eqelr 2373 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  ( T `  y ) )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3018cnmptid 17351 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  y )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3110hhnv 21740 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3210hhvs 21745 . . . . . . . . . . 11  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
3311, 7, 32vmcn 21266 . . . . . . . . . 10  |-  ( <. <.  +h  ,  .h  >. , 
normh >.  e.  NrmCVec  ->  -h  e.  ( ( ( MetOpen `  ( normh  o.  -h  )
)  tX  ( MetOpen `  ( normh  o.  -h  )
) )  Cn  ( MetOpen
`  ( normh  o.  -h  ) ) ) )
3431, 33mp1i 11 . . . . . . . . 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 17356 . . . . . . . 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 17029 . . . . . . 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 443 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f : NN --> ran  T )
384shssii 21788 . . . . . . . . . . . . . 14  |-  ran  T  C_ 
~H
39 fss 5363 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  ran  T  C_  ~H )  ->  f : NN --> ~H )
4037, 38, 39sylancl 643 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f : NN --> ~H )
41 ffvelrn 5625 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  e.  ~H )
4240, 41sylan 457 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ~H )
43 fveq2 5486 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  ( T `  y )  =  ( T `  ( f `  k
) ) )
44 id 19 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  y  =  ( f `  k ) )
4543, 44oveq12d 5838 . . . . . . . . . . . . 13  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 ( f `  k ) )  -h  ( f `  k
) ) )
46 eqid 2284 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  =  ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )
47 ovex 5845 . . . . . . . . . . . . 13  |-  ( ( T `  ( f `
 k ) )  -h  ( f `  k ) )  e. 
_V
4845, 46, 47fvmpt 5564 . . . . . . . . . . . 12  |-  ( ( f `  k )  e.  ~H  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  (
f `  k )
)  =  ( ( T `  ( f `
 k ) )  -h  ( f `  k ) ) )
4942, 48syl 15 . . . . . . . . . . 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 ) ) )
50 ffn 5355 . . . . . . . . . . . . . . . 16  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
5119, 50ax-mp 8 . . . . . . . . . . . . . . 15  |-  T  Fn  ~H
52 fveq2 5486 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  ( T `  y )  =  ( T `  ( T `  x ) ) )
53 id 19 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  y  =  ( T `  x ) )
5452, 53eqeq12d 2298 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( T `  x )  ->  (
( T `  y
)  =  y  <->  ( T `  ( T `  x
) )  =  ( T `  x ) ) )
5554ralrn 5630 . . . . . . . . . . . . . . 15  |-  ( T  Fn  ~H  ->  ( A. y  e.  ran  T ( T `  y
)  =  y  <->  A. x  e.  ~H  ( T `  ( T `  x ) )  =  ( T `
 x ) ) )
5651, 55ax-mp 8 . . . . . . . . . . . . . 14  |-  ( A. y  e.  ran  T ( T `  y )  =  y  <->  A. x  e.  ~H  ( T `  ( T `  x ) )  =  ( T `
 x ) )
57 hmopidmch.2 . . . . . . . . . . . . . . . 16  |-  ( T  o.  T )  =  T
5857fveq1i 5487 . . . . . . . . . . . . . . 15  |-  ( ( T  o.  T ) `
 x )  =  ( T `  x
)
5919, 19hocoi 22340 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( T  o.  T
) `  x )  =  ( T `  ( T `  x ) ) )
6058, 59syl5reqr 2331 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( T `  ( T `  x ) )  =  ( T `  x
) )
6156, 60mprgbir 2614 . . . . . . . . . . . . 13  |-  A. y  e.  ran  T ( T `
 y )  =  y
62 ffvelrn 5625 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  k  e.  NN )  ->  ( f `  k )  e.  ran  T )
6362adantlr 695 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ran  T )
6443, 44eqeq12d 2298 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  =  y  <->  ( T `  ( f `  k
) )  =  ( f `  k ) ) )
6564rspccv 2882 . . . . . . . . . . . . 13  |-  ( A. y  e.  ran  T ( T `  y )  =  y  ->  (
( f `  k
)  e.  ran  T  ->  ( T `  (
f `  k )
)  =  ( f `
 k ) ) )
6661, 63, 65mpsyl 59 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  ( T `  ( f `  k ) )  =  ( f `  k
) )
6766, 42eqeltrd 2358 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  ( T `  ( f `  k ) )  e. 
~H )
68 hvsubeq0 21643 . . . . . . . . . . . . 13  |-  ( ( ( T `  (
f `  k )
)  e.  ~H  /\  ( f `  k
)  e.  ~H )  ->  ( ( ( T `
 ( f `  k ) )  -h  ( f `  k
) )  =  0h  <->  ( T `  ( f `
 k ) )  =  ( f `  k ) ) )
6967, 42, 68syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( ( T `  ( f `  k
) )  -h  (
f `  k )
)  =  0h  <->  ( T `  ( f `  k
) )  =  ( f `  k ) ) )
7066, 69mpbird 223 . . . . . . . . . . 11  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( T `  (
f `  k )
)  -h  ( f `
 k ) )  =  0h )
7149, 70eqtrd 2316 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  (
f `  k )
)  =  0h )
72 fvco3 5558 . . . . . . . . . . 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
) ) )
7372adantlr 695 . . . . . . . . . 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 ) ) )
74 ax-hv0cl 21579 . . . . . . . . . . . . 13  |-  0h  e.  ~H
7574elexi 2798 . . . . . . . . . . . 12  |-  0h  e.  _V
7675fvconst2 5691 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
( NN  X.  { 0h } ) `  k
)  =  0h )
7776adantl 452 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( NN  X.  { 0h } ) `  k
)  =  0h )
7871, 73, 773eqtr4d 2326 . . . . . . . . 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
) )
7978ralrimiva 2627 . . . . . . . 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
) )
80 ovex 5845 . . . . . . . . . . 11  |-  ( ( T `  y )  -h  y )  e. 
_V
8180, 46fnmpti 5338 . . . . . . . . . 10  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  Fn  ~H
82 fnfco 5373 . . . . . . . . . 10  |-  ( ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y
) )  Fn  ~H  /\  f : NN --> ~H )  ->  ( ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )  o.  f )  Fn  NN )
8381, 40, 82sylancr 644 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f )  Fn  NN )
8475fconst 5393 . . . . . . . . . 10  |-  ( NN 
X.  { 0h }
) : NN --> { 0h }
85 ffn 5355 . . . . . . . . . 10  |-  ( ( NN  X.  { 0h } ) : NN --> { 0h }  ->  ( NN  X.  { 0h }
)  Fn  NN )
8684, 85ax-mp 8 . . . . . . . . 9  |-  ( NN 
X.  { 0h }
)  Fn  NN
87 eqfnfv 5584 . . . . . . . . 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 ) ) )
8883, 86, 87sylancl 643 . . . . . . . 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 ) ) )
8979, 88mpbird 223 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f )  =  ( NN  X.  { 0h } ) )
90 vex 2792 . . . . . . . . . 10  |-  x  e. 
_V
9190hlimveci 21765 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  x  e.  ~H )
9291adantl 452 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ~H )
93 fveq2 5486 . . . . . . . . . 10  |-  ( y  =  x  ->  ( T `  y )  =  ( T `  x ) )
94 id 19 . . . . . . . . . 10  |-  ( y  =  x  ->  y  =  x )
9593, 94oveq12d 5838 . . . . . . . . 9  |-  ( y  =  x  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 x )  -h  x ) )
96 ovex 5845 . . . . . . . . 9  |-  ( ( T `  x )  -h  x )  e. 
_V
9795, 46, 96fvmpt 5564 . . . . . . . 8  |-  ( x  e.  ~H  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  x
)  =  ( ( T `  x )  -h  x ) )
9892, 97syl 15 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) ) `
 x )  =  ( ( T `  x )  -h  x
) )
9936, 89, 983brtr3d 4053 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) ) ( ( T `  x )  -h  x ) )
10074a1i 10 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  0h  e.  ~H )
101 1z 10049 . . . . . . . 8  |-  1  e.  ZZ
102101a1i 10 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  1  e.  ZZ )
103 nnuz 10259 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
104103lmconst 16987 . . . . . . 7  |-  ( ( ( MetOpen `  ( normh  o. 
-h  ) )  e.  (TopOn `  ~H )  /\  0h  e.  ~H  /\  1  e.  ZZ )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) 0h )
10518, 100, 102, 104syl3anc 1182 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) ) 0h )
1069, 99, 105lmmo 17104 . . . . 5  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( T `
 x )  -h  x )  =  0h )
10719ffvelrni 5626 . . . . . . 7  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
10892, 107syl 15 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  e.  ~H )
109 hvsubeq0 21643 . . . . . 6  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( ( T `
 x )  -h  x )  =  0h  <->  ( T `  x )  =  x ) )
110108, 92, 109syl2anc 642 . . . . 5  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( ( T `  x )  -h  x )  =  0h  <->  ( T `  x )  =  x ) )
111106, 110mpbid 201 . . . 4  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  =  x )
112 fnfvelrn 5624 . . . . 5  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ran  T
)
11351, 92, 112sylancr 644 . . . 4  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  e.  ran  T )
114111, 113eqeltrrd 2359 . . 3  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
115114gen2 1534 . 2  |-  A. f A. x ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
116 isch2 21799 . 2  |-  ( ran 
T  e.  CH  <->  ( ran  T  e.  SH  /\  A. f A. x ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T ) ) )
1174, 115, 116mpbir2an 886 1  |-  ran  T  e.  CH
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1685   A.wral 2544    C_ wss 3153   {csn 3641   <.cop 3644   class class class wbr 4024    e. cmpt 4078    X. cxp 4686   ran crn 4689    |` cres 4690    o. ccom 4692    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5820    ^m cmap 6768   1c1 8734   NNcn 9742   ZZcz 10020   * Metcxmt 16365   MetOpencmopn 16368  TopOnctopon 16628    Cn ccn 16950   ~~> tclm 16952   Hauscha 17032    tX ctx 17251   NrmCVeccnv 21134   ~Hchil 21495    +h cva 21496    .h csm 21497   normhcno 21499   0hc0v 21500    -h cmv 21501    ~~>v chli 21503   SHcsh 21504   CHcch 21505   ConOpccop 21522   LinOpclo 21523   BndLinOpcbo 21524   HrmOpcho 21526
This theorem is referenced by:  hmopidmpji  22728  hmopidmch  22729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cc 8057  ax-dc 8068  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813  ax-hilex 21575  ax-hfvadd 21576  ax-hvcom 21577  ax-hvass 21578  ax-hv0cl 21579  ax-hvaddid 21580  ax-hfvmul 21581  ax-hvmulid 21582  ax-hvmulass 21583  ax-hvdistr1 21584  ax-hvdistr2 21585  ax-hvmul0 21586  ax-hfi 21654  ax-his1 21657  ax-his2 21658  ax-his3 21659  ax-his4 21660  ax-hcompl 21777
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-omul 6480  df-er 6656  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-acn 7571  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10656  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-rlim 11959  df-sum 12155  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-cn 16953  df-cnp 16954  df-lm 16955  df-t1 17038  df-haus 17039  df-cmp 17110  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-fcls 17632  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-cfil 18677  df-cau 18678  df-cmet 18679  df-grpo 20852  df-gid 20853  df-ginv 20854  df-gdiv 20855  df-ablo 20943  df-subgo 20963  df-vc 21096  df-nv 21142  df-va 21145  df-ba 21146  df-sm 21147  df-0v 21148  df-vs 21149  df-nmcv 21150  df-ims 21151  df-dip 21268  df-ssp 21292  df-lno 21316  df-nmoo 21317  df-blo 21318  df-0o 21319  df-ph 21385  df-cbn 21436  df-hlo 21459  df-hnorm 21544  df-hba 21545  df-hvsub 21547  df-hlim 21548  df-hcau 21549  df-sh 21782  df-ch 21797  df-oc 21827  df-ch0 21828  df-shs 21883  df-pjh 21970  df-h0op 22324  df-nmop 22415  df-cnop 22416  df-lnop 22417  df-bdop 22418  df-unop 22419  df-hmop 22420
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