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Theorem hmopidmchi 25474
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 25265 . . . 4  |-  ( T  e.  HrmOp  ->  T  e.  LinOp
)
31, 2ax-mp 5 . . 3  |-  T  e. 
LinOp
43rnelshi 25382 . 2  |-  ran  T  e.  SH
5 eqid 2441 . . . . . . . 8  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
65hilxmet 24516 . . . . . . 7  |-  ( normh  o. 
-h  )  e.  ( *Met `  ~H )
7 eqid 2441 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
87methaus 19995 . . . . . . 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 2441 . . . . . . . . . . . 12  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
1110, 5hhims 24493 . . . . . . . . . . . 12  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
1210, 11, 7hhlm 24520 . . . . . . . . . . 11  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
13 resss 5131 . . . . . . . . . . 11  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1412, 13eqsstri 3383 . . . . . . . . . 10  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1514ssbri 4331 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  f ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) ) x )
1615adantl 463 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) x )
177mopntopon 19914 . . . . . . . . . 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 25292 . . . . . . . . . . . 12  |-  T : ~H
--> ~H
2019a1i 11 . . . . . . . . . . 11  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T : ~H --> ~H )
2120feqmptd 5741 . . . . . . . . . 10  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  T  =  ( y  e.  ~H  |->  ( T `  y ) ) )
22 hmopbdoptHIL 25311 . . . . . . . . . . . . 13  |-  ( T  e.  HrmOp  ->  T  e.  BndLinOp )
231, 22ax-mp 5 . . . . . . . . . . . 12  |-  T  e.  BndLinOp
24 lnopcnbd 25359 . . . . . . . . . . . . 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 25227 . . . . . . . . . . 11  |-  ConOp  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( MetOpen `  ( normh  o. 
-h  ) ) )
2826, 27eleqtri 2513 . . . . . . . . . 10  |-  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) )
2921, 28syl6eqelr 2530 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  ( T `  y ) )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3018cnmptid 19134 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( y  e. 
~H  |->  y )  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( MetOpen `  ( normh  o.  -h  ) ) ) )
3110hhnv 24486 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  e.  NrmCVec
3210hhvs 24491 . . . . . . . . . . 11  |-  -h  =  ( -v `  <. <.  +h  ,  .h  >. ,  normh >. )
3311, 7, 32vmcn 24013 . . . . . . . . . 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 19139 . . . . . . . 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 18809 . . . . . . 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 454 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f : NN --> ran  T )
384shssii 24534 . . . . . . . . . . . . . 14  |-  ran  T  C_ 
~H
39 fss 5564 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  ran  T  C_  ~H )  ->  f : NN --> ~H )
4037, 38, 39sylancl 657 . . . . . . . . . . . . 13  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  f : NN --> ~H )
4140ffvelrnda 5840 . . . . . . . . . . . 12  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ~H )
42 fveq2 5688 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  ( T `  y )  =  ( T `  ( f `  k
) ) )
43 id 22 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  y  =  ( f `  k ) )
4442, 43oveq12d 6108 . . . . . . . . . . . . 13  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 ( f `  k ) )  -h  ( f `  k
) ) )
45 eqid 2441 . . . . . . . . . . . . 13  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  =  ( y  e. 
~H  |->  ( ( T `
 y )  -h  y ) )
46 ovex 6115 . . . . . . . . . . . . 13  |-  ( ( T `  ( f `
 k ) )  -h  ( f `  k ) )  e. 
_V
4744, 45, 46fvmpt 5771 . . . . . . . . . . . 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 5556 . . . . . . . . . . . . . . . 16  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
5019, 49ax-mp 5 . . . . . . . . . . . . . . 15  |-  T  Fn  ~H
51 fveq2 5688 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  ( T `  y )  =  ( T `  ( T `  x ) ) )
52 id 22 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( T `  x )  ->  y  =  ( T `  x ) )
5351, 52eqeq12d 2455 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( T `  x )  ->  (
( T `  y
)  =  y  <->  ( T `  ( T `  x
) )  =  ( T `  x ) ) )
5453ralrn 5843 . . . . . . . . . . . . . . 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 5689 . . . . . . . . . . . . . . 15  |-  ( ( T  o.  T ) `
 x )  =  ( T `  x
)
5819, 19hocoi 25087 . . . . . . . . . . . . . . 15  |-  ( x  e.  ~H  ->  (
( T  o.  T
) `  x )  =  ( T `  ( T `  x ) ) )
5957, 58syl5reqr 2488 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  ( T `  ( T `  x ) )  =  ( T `  x
) )
6055, 59mprgbir 2784 . . . . . . . . . . . . 13  |-  A. y  e.  ran  T ( T `
 y )  =  y
61 ffvelrn 5838 . . . . . . . . . . . . . 14  |-  ( ( f : NN --> ran  T  /\  k  e.  NN )  ->  ( f `  k )  e.  ran  T )
6261adantlr 709 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
f `  k )  e.  ran  T )
6342, 43eqeq12d 2455 . . . . . . . . . . . . . 14  |-  ( y  =  ( f `  k )  ->  (
( T `  y
)  =  y  <->  ( T `  ( f `  k
) )  =  ( f `  k ) ) )
6463rspccv 3067 . . . . . . . . . . . . 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 2515 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  ( T `  ( f `  k ) )  e. 
~H )
67 hvsubeq0 24389 . . . . . . . . . . . . 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 656 . . . . . . . . . . . 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 2473 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( y  e.  ~H  |->  ( ( T `  y )  -h  y
) ) `  (
f `  k )
)  =  0h )
71 fvco3 5765 . . . . . . . . . . 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 709 . . . . . . . . . 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 24324 . . . . . . . . . . . . 13  |-  0h  e.  ~H
7473elexi 2980 . . . . . . . . . . . 12  |-  0h  e.  _V
7574fvconst2 5930 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
( NN  X.  { 0h } ) `  k
)  =  0h )
7675adantl 463 . . . . . . . . . 10  |-  ( ( ( f : NN --> ran  T  /\  f  ~~>v  x )  /\  k  e.  NN )  ->  (
( NN  X.  { 0h } ) `  k
)  =  0h )
7770, 72, 763eqtr4d 2483 . . . . . . . . 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 2797 . . . . . . . 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 6115 . . . . . . . . . . 11  |-  ( ( T `  y )  -h  y )  e. 
_V
8079, 45fnmpti 5536 . . . . . . . . . 10  |-  ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  Fn  ~H
81 fnfco 5574 . . . . . . . . . 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 658 . . . . . . . . 9  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( y  e.  ~H  |->  ( ( T `  y )  -h  y ) )  o.  f )  Fn  NN )
8374fconst 5593 . . . . . . . . . 10  |-  ( NN 
X.  { 0h }
) : NN --> { 0h }
84 ffn 5556 . . . . . . . . . 10  |-  ( ( NN  X.  { 0h } ) : NN --> { 0h }  ->  ( NN  X.  { 0h }
)  Fn  NN )
8583, 84ax-mp 5 . . . . . . . . 9  |-  ( NN 
X.  { 0h }
)  Fn  NN
86 eqfnfv 5794 . . . . . . . . 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 657 . . . . . . . 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 2973 . . . . . . . . . 10  |-  x  e. 
_V
9089hlimveci 24511 . . . . . . . . 9  |-  ( f 
~~>v  x  ->  x  e.  ~H )
9190adantl 463 . . . . . . . 8  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ~H )
92 fveq2 5688 . . . . . . . . . 10  |-  ( y  =  x  ->  ( T `  y )  =  ( T `  x ) )
93 id 22 . . . . . . . . . 10  |-  ( y  =  x  ->  y  =  x )
9492, 93oveq12d 6108 . . . . . . . . 9  |-  ( y  =  x  ->  (
( T `  y
)  -h  y )  =  ( ( T `
 x )  -h  x ) )
95 ovex 6115 . . . . . . . . 9  |-  ( ( T `  x )  -h  x )  e. 
_V
9694, 45, 95fvmpt 5771 . . . . . . . 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 4318 . . . . . 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 10673 . . . . . . 7  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  1  e.  ZZ )
101 nnuz 10892 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
102101lmconst 18765 . . . . . . 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 1213 . . . . . 6  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( NN  X.  { 0h } ) ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) ) 0h )
1049, 98, 103lmmo 18884 . . . . 5  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( ( T `
 x )  -h  x )  =  0h )
10519ffvelrni 5839 . . . . . . 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 24389 . . . . . 6  |-  ( ( ( T `  x
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( ( T `
 x )  -h  x )  =  0h  <->  ( T `  x )  =  x ) )
108106, 91, 107syl2anc 656 . . . . 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 5837 . . . . 5  |-  ( ( T  Fn  ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ran  T
)
11150, 91, 110sylancr 658 . . . 4  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  ( T `  x )  e.  ran  T )
112109, 111eqeltrrd 2516 . . 3  |-  ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
113112gen2 1597 . 2  |-  A. f A. x ( ( f : NN --> ran  T  /\  f  ~~>v  x )  ->  x  e.  ran  T )
114 isch2 24545 . 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 906 1  |-  ran  T  e.  CH
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1362    = wceq 1364    e. wcel 1761   A.wral 2713    C_ wss 3325   {csn 3874   <.cop 3880   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   ran crn 4837    |` cres 4838    o. ccom 4840    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^m cmap 7210   1c1 9279   NNcn 10318   ZZcz 10642   *Metcxmt 17701   MetOpencmopn 17706  TopOnctopon 18399    Cn ccn 18728   ~~> tclm 18730   Hauscha 18812    tX ctx 19033   NrmCVeccnv 23881   ~Hchil 24240    +h cva 24241    .h csm 24242   normhcno 24244   0hc0v 24245    -h cmv 24246    ~~>v chli 24248   SHcsh 24249   CHcch 24250   ConOpccop 24267   LinOpclo 24268   BndLinOpcbo 24269   HrmOpcho 24271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cc 8600  ax-dc 8611  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358  ax-hilex 24320  ax-hfvadd 24321  ax-hvcom 24322  ax-hvass 24323  ax-hv0cl 24324  ax-hvaddid 24325  ax-hfvmul 24326  ax-hvmulid 24327  ax-hvmulass 24328  ax-hvdistr1 24329  ax-hvdistr2 24330  ax-hvmul0 24331  ax-hfi 24400  ax-his1 24403  ax-his2 24404  ax-his3 24405  ax-his4 24406  ax-hcompl 24523
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-cn 18731  df-cnp 18732  df-lm 18733  df-t1 18818  df-haus 18819  df-cmp 18890  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-fcls 19414  df-xms 19795  df-ms 19796  df-tms 19797  df-cncf 20354  df-cfil 20666  df-cau 20667  df-cmet 20668  df-grpo 23597  df-gid 23598  df-ginv 23599  df-gdiv 23600  df-ablo 23688  df-subgo 23708  df-vc 23843  df-nv 23889  df-va 23892  df-ba 23893  df-sm 23894  df-0v 23895  df-vs 23896  df-nmcv 23897  df-ims 23898  df-dip 24015  df-ssp 24039  df-lno 24063  df-nmoo 24064  df-blo 24065  df-0o 24066  df-ph 24132  df-cbn 24183  df-hlo 24206  df-hnorm 24289  df-hba 24290  df-hvsub 24292  df-hlim 24293  df-hcau 24294  df-sh 24528  df-ch 24543  df-oc 24574  df-ch0 24575  df-shs 24630  df-pjh 24717  df-h0op 25071  df-nmop 25162  df-cnop 25163  df-lnop 25164  df-bdop 25165  df-unop 25166  df-hmop 25167
This theorem is referenced by:  hmopidmpji  25475  hmopidmch  25476
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