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Theorem sitgclg 24609
Description: Closure of the Bochner integral on a simple functions. This version is very generic, thus the many hypothesis. See sitgclbn 24610 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.)
Hypotheses
Ref Expression
sitgval.b  |-  B  =  ( Base `  W
)
sitgval.j  |-  J  =  ( TopOpen `  W )
sitgval.s  |-  S  =  (sigaGen `  J )
sitgval.0  |-  .0.  =  ( 0g `  W )
sitgval.x  |-  .x.  =  ( .s `  W )
sitgval.h  |-  H  =  (RRHom `  (Scalar `  W
) )
sitgval.1  |-  ( ph  ->  W  e.  V )
sitgval.2  |-  ( ph  ->  M  e.  U. ran measures )
sibfmbl.1  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
sitgclg.g  |-  G  =  (Scalar `  W )
sitgclg.d  |-  D  =  ( ( dist `  G
)  |`  ( ( Base `  G )  X.  ( Base `  G ) ) )
sitgclg.1  |-  ( ph  ->  W  e.  TopSp )
sitgclg.2  |-  ( ph  ->  W  e. CMnd )
sitgclg.3  |-  ( ph  ->  G  e.  DivRing )
sitgclg.4  |-  ( ph  ->  G  e. NrmRing )
sitgclg.5  |-  ( ph  ->  ( ZMod `  G
)  e. NrmMod )
sitgclg.6  |-  ( ph  ->  (chr `  G )  =  0 )
sitgclg.7  |-  ( ph  ->  G  e.  TopSp )
sitgclg.8  |-  ( ph  ->  G  e. CUnifSp )
sitgclg.9  |-  ( ph  ->  ( TopOpen `  G )  e.  Haus )
sitgclg.10  |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
)
sitgclg.11  |-  ( (
ph  /\  m  e.  ( H " ( 0 [,)  +oo ) )  /\  x  e.  B )  ->  ( m  .x.  x
)  e.  B )
Assertion
Ref Expression
sitgclg  |-  ( ph  ->  ( ( Wsitg M
) `  F )  e.  B )
Distinct variable groups:    B, m    x, F    m, H    x, m, M    S, m    m, W, x    .0. , m, x    .x. , m    ph, x    x, B    m, F    m, G    ph, m
Allowed substitution hints:    D( x, m)    S( x)    .x. ( x)    G( x)    H( x)    J( x, m)    V( x, m)

Proof of Theorem sitgclg
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitgval.b . . 3  |-  B  =  ( Base `  W
)
2 sitgval.j . . 3  |-  J  =  ( TopOpen `  W )
3 sitgval.s . . 3  |-  S  =  (sigaGen `  J )
4 sitgval.0 . . 3  |-  .0.  =  ( 0g `  W )
5 sitgval.x . . 3  |-  .x.  =  ( .s `  W )
6 sitgval.h . . 3  |-  H  =  (RRHom `  (Scalar `  W
) )
7 sitgval.1 . . 3  |-  ( ph  ->  W  e.  V )
8 sitgval.2 . . 3  |-  ( ph  ->  M  e.  U. ran measures )
9 sibfmbl.1 . . 3  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
101, 2, 3, 4, 5, 6, 7, 8, 9sitgfval 24608 . 2  |-  ( ph  ->  ( ( Wsitg M
) `  F )  =  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) ) )
11 sitgclg.2 . . 3  |-  ( ph  ->  W  e. CMnd )
12 rnexg 5090 . . . 4  |-  ( F  e.  dom  ( Wsitg M )  ->  ran  F  e.  _V )
13 difexg 4311 . . . 4  |-  ( ran 
F  e.  _V  ->  ( ran  F  \  {  .0.  } )  e.  _V )
149, 12, 133syl 19 . . 3  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  e. 
_V )
15 simpl 444 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  ph )
161, 2, 3, 4, 5, 6, 7, 8, 9sibfima 24606 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) )
17 sitgclg.d . . . . . . . . . . . 12  |-  D  =  ( ( dist `  G
)  |`  ( ( Base `  G )  X.  ( Base `  G ) ) )
18 sitgclg.g . . . . . . . . . . . . . 14  |-  G  =  (Scalar `  W )
1918fveq2i 5690 . . . . . . . . . . . . 13  |-  ( dist `  G )  =  (
dist `  (Scalar `  W
) )
2018fveq2i 5690 . . . . . . . . . . . . . 14  |-  ( Base `  G )  =  (
Base `  (Scalar `  W
) )
2120, 20xpeq12i 4859 . . . . . . . . . . . . 13  |-  ( (
Base `  G )  X.  ( Base `  G
) )  =  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) )
2219, 21reseq12i 5103 . . . . . . . . . . . 12  |-  ( (
dist `  G )  |`  ( ( Base `  G
)  X.  ( Base `  G ) ) )  =  ( ( dist `  (Scalar `  W )
)  |`  ( ( Base `  (Scalar `  W )
)  X.  ( Base `  (Scalar `  W )
) ) )
2317, 22eqtri 2424 . . . . . . . . . . 11  |-  D  =  ( ( dist `  (Scalar `  W ) )  |`  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) ) )
24 eqid 2404 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
25 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2618fveq2i 5690 . . . . . . . . . . 11  |-  ( TopOpen `  G )  =  (
TopOpen `  (Scalar `  W
) )
27 sitgclg.3 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  DivRing )
2818, 27syl5eqelr 2489 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e.  DivRing )
29 sitgclg.4 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. NrmRing )
3018, 29syl5eqelr 2489 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e. NrmRing )
3118fveq2i 5690 . . . . . . . . . . . 12  |-  ( ZMod
`  G )  =  ( ZMod `  (Scalar `  W ) )
32 sitgclg.5 . . . . . . . . . . . 12  |-  ( ph  ->  ( ZMod `  G
)  e. NrmMod )
3331, 32syl5eqelr 2489 . . . . . . . . . . 11  |-  ( ph  ->  ( ZMod `  (Scalar `  W ) )  e. NrmMod
)
3418fveq2i 5690 . . . . . . . . . . . 12  |-  (chr `  G )  =  (chr
`  (Scalar `  W )
)
35 sitgclg.6 . . . . . . . . . . . 12  |-  ( ph  ->  (chr `  G )  =  0 )
3634, 35syl5eqr 2450 . . . . . . . . . . 11  |-  ( ph  ->  (chr `  (Scalar `  W
) )  =  0 )
37 sitgclg.7 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  TopSp )
3818, 37syl5eqelr 2489 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e.  TopSp )
39 sitgclg.8 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. CUnifSp )
4018, 39syl5eqelr 2489 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e. CUnifSp )
41 sitgclg.9 . . . . . . . . . . 11  |-  ( ph  ->  ( TopOpen `  G )  e.  Haus )
4218fveq2i 5690 . . . . . . . . . . . 12  |-  (UnifSt `  G )  =  (UnifSt `  (Scalar `  W )
)
43 sitgclg.10 . . . . . . . . . . . 12  |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
)
4442, 43syl5eqr 2450 . . . . . . . . . . 11  |-  ( ph  ->  (UnifSt `  (Scalar `  W
) )  =  (metUnif `  D ) )
4523, 24, 25, 26, 28, 30, 33, 36, 38, 40, 41, 44rrhf 24334 . . . . . . . . . 10  |-  ( ph  ->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
466feq1i 5544 . . . . . . . . . 10  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  <->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
4745, 46sylibr 204 . . . . . . . . 9  |-  ( ph  ->  H : RR --> ( Base `  (Scalar `  W )
) )
48 ffun 5552 . . . . . . . . 9  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  Fun  H )
4947, 48syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  H )
50 0re 9047 . . . . . . . . . . 11  |-  0  e.  RR
51 pnfxr 10669 . . . . . . . . . . 11  |-  +oo  e.  RR*
52 icossre 10947 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
5350, 51, 52mp2an 654 . . . . . . . . . 10  |-  ( 0 [,)  +oo )  C_  RR
5453a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( 0 [,)  +oo )  C_  RR )
55 fdm 5554 . . . . . . . . . 10  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  dom  H  =  RR )
5647, 55syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  H  =  RR )
5754, 56sseqtr4d 3345 . . . . . . . 8  |-  ( ph  ->  ( 0 [,)  +oo )  C_  dom  H )
58 funfvima2 5933 . . . . . . . 8  |-  ( ( Fun  H  /\  (
0 [,)  +oo )  C_  dom  H )  ->  (
( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,)  +oo ) ) ) )
5949, 57, 58syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,)  +oo ) ) ) )
6059imp 419 . . . . . 6  |-  ( (
ph  /\  ( M `  ( `' F " { x } ) )  e.  ( 0 [,)  +oo ) )  -> 
( H `  ( M `  ( `' F " { x }
) ) )  e.  ( H " (
0 [,)  +oo ) ) )
6115, 16, 60syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( H `  ( M `  ( `' F " { x }
) ) )  e.  ( H " (
0 [,)  +oo ) ) )
62 isrnmeas 24507 . . . . . . . . . . . . 13  |-  ( M  e.  U. ran measures  ->  ( dom  M  e.  U. ran sigAlgebra  /\  ( M : dom  M --> ( 0 [,]  +oo )  /\  ( M `  (/) )  =  0  /\ 
A. y  e.  ~P  dom  M ( ( y  ~<_  om  /\ Disj  z  e.  y z )  ->  ( M `  U. y )  = Σ* z  e.  y ( M `  z ) ) ) ) )
6362simpld 446 . . . . . . . . . . . 12  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
648, 63syl 16 . . . . . . . . . . 11  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
65 fvex 5701 . . . . . . . . . . . . . . 15  |-  ( TopOpen `  W )  e.  _V
662, 65eqeltri 2474 . . . . . . . . . . . . . 14  |-  J  e. 
_V
6766a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  _V )
6867sgsiga 24478 . . . . . . . . . . . 12  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
693, 68syl5eqel 2488 . . . . . . . . . . 11  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
701, 2, 3, 4, 5, 6, 7, 8, 9sibfmbl 24603 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
7164, 69, 70mbfmf 24558 . . . . . . . . . 10  |-  ( ph  ->  F : U. dom  M --> U. S )
72 frn 5556 . . . . . . . . . 10  |-  ( F : U. dom  M --> U. S  ->  ran  F  C_ 
U. S )
7371, 72syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  U. S
)
743unieqi 3985 . . . . . . . . . . 11  |-  U. S  =  U. (sigaGen `  J
)
75 unisg 24479 . . . . . . . . . . . 12  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
7666, 75mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
7774, 76syl5eq 2448 . . . . . . . . . 10  |-  ( ph  ->  U. S  =  U. J )
78 sitgclg.1 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  TopSp )
791, 2tpsuni 16958 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
8078, 79syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
8177, 80eqtr4d 2439 . . . . . . . . 9  |-  ( ph  ->  U. S  =  B )
8273, 81sseqtrd 3344 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  B
)
8382ssdifd 3443 . . . . . . 7  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  C_  ( B  \  {  .0.  } ) )
8483sselda 3308 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  ( B  \  {  .0.  } ) )
8584eldifad 3292 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  B )
86 simp2 958 . . . . . 6  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,)  +oo ) )  /\  x  e.  B )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H " ( 0 [,)  +oo ) ) )
87 eleq1 2464 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  e.  ( H " (
0 [,)  +oo ) )  <-> 
( H `  ( M `  ( `' F " { x }
) ) )  e.  ( H " (
0 [,)  +oo ) ) ) )
88873anbi2d 1259 . . . . . . . 8  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( ( ph  /\  m  e.  ( H
" ( 0 [,) 
+oo ) )  /\  x  e.  B )  <->  (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,)  +oo ) )  /\  x  e.  B ) ) )
89 oveq1 6047 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
9089eleq1d 2470 . . . . . . . 8  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( ( m 
.x.  x )  e.  B  <->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B ) )
9188, 90imbi12d 312 . . . . . . 7  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( ( (
ph  /\  m  e.  ( H " ( 0 [,)  +oo ) )  /\  x  e.  B )  ->  ( m  .x.  x
)  e.  B )  <-> 
( ( ph  /\  ( H `  ( M `
 ( `' F " { x } ) ) )  e.  ( H " ( 0 [,)  +oo ) )  /\  x  e.  B )  ->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B ) ) )
92 sitgclg.11 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( H " ( 0 [,)  +oo ) )  /\  x  e.  B )  ->  ( m  .x.  x
)  e.  B )
9391, 92vtoclg 2971 . . . . . 6  |-  ( ( H `  ( M `
 ( `' F " { x } ) ) )  e.  ( H " ( 0 [,)  +oo ) )  -> 
( ( ph  /\  ( H `  ( M `
 ( `' F " { x } ) ) )  e.  ( H " ( 0 [,)  +oo ) )  /\  x  e.  B )  ->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B ) )
9486, 93mpcom 34 . . . . 5  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,)  +oo ) )  /\  x  e.  B )  ->  (
( H `  ( M `  ( `' F " { x }
) ) )  .x.  x )  e.  B
)
9515, 61, 85, 94syl3anc 1184 . . . 4  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B )
96 eqid 2404 . . . 4  |-  ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )  =  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
9795, 96fmptd 5852 . . 3  |-  ( ph  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) : ( ran  F  \  {  .0.  } ) --> B )
981, 2, 3, 4, 5, 6, 7, 8, 9sibfrn 24605 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
99 cnvimass 5183 . . . . . . 7  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  dom  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
10096dmmptss 5325 . . . . . . 7  |-  dom  (
x  e.  ( ran 
F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) 
C_  ( ran  F  \  {  .0.  } )
10199, 100sstri 3317 . . . . . 6  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ( ran  F  \  {  .0.  } )
102 difss 3434 . . . . . 6  |-  ( ran 
F  \  {  .0.  } )  C_  ran  F
103101, 102sstri 3317 . . . . 5  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ran  F
104103a1i 11 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) ) 
C_  ran  F )
105 ssfi 7288 . . . 4  |-  ( ( ran  F  e.  Fin  /\  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) ) 
C_  ran  F )  ->  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
10698, 104, 105syl2anc 643 . . 3  |-  ( ph  ->  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
1071, 4, 11, 14, 97, 106gsumcl 15476 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) )  e.  B )
10810, 107eqeltrd 2478 1  |-  ( ph  ->  ( ( Wsitg M
) `  F )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   U.cuni 3975  Disj wdisj 4142   class class class wbr 4172    e. cmpt 4226   omcom 4804    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040    ~<_ cdom 7066   Fincfn 7068   RRcr 8945   0cc0 8946    +oocpnf 9073   RR*cxr 9075   (,)cioo 10872   [,)cico 10874   [,]cicc 10875   Basecbs 13424  Scalarcsca 13487   .scvsca 13488   distcds 13493   TopOpenctopn 13604   topGenctg 13620   0gc0g 13678    gsumg cgsu 13679  CMndccmn 15367   DivRingcdr 15790  metUnifcmetu 16648   ZModczlm 16734  chrcchr 16735   TopSpctps 16916   Hauscha 17326  UnifStcuss 18236  CUnifSpccusp 18280  NrmRingcnrg 18580  NrmModcnlm 18581  RRHomcrrh 24330  Σ*cesum 24377  sigAlgebracsiga 24443  sigaGencsigagen 24474  measurescmeas 24502  sitgcsitg 24597
This theorem is referenced by:  sitgclbn  24610  sitmcl  24616
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-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
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-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  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-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-mod 11206  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-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083  df-gz 13253  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-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-od 15122  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-subrg 15821  df-abv 15860  df-lmod 15907  df-nzr 16284  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-metu 16657  df-cnfld 16659  df-zrh 16737  df-zlm 16738  df-chr 16739  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-haus 17333  df-reg 17334  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-cnext 18044  df-ust 18183  df-utop 18214  df-uss 18239  df-usp 18240  df-ucn 18259  df-cfilu 18270  df-cusp 18281  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-cncf 18861  df-cfil 19161  df-cmet 19163  df-cms 19241  df-qqh 24310  df-rrh 24331  df-esum 24378  df-siga 24444  df-sigagen 24475  df-meas 24503  df-mbfm 24554  df-sitg 24598
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