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Theorem sitgclg 28157
Description: Closure of the Bochner integral on simple functions, generic version. See sitgclbn 28158 for the version for Banach spaces. (Contributed by Thierry Arnoux, 24-Feb-2018.) (Proof shortened by AV, 12-Dec-2019.)
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  ->  (Scalar `  W )  e. ℝExt  )
sitgclg.4  |-  ( (
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
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 28156 . 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 6717 . . . 4  |-  ( F  e.  dom  ( Wsitg M )  ->  ran  F  e.  _V )
13 difexg 4585 . . . 4  |-  ( ran 
F  e.  _V  ->  ( ran  F  \  {  .0.  } )  e.  _V )
149, 12, 133syl 20 . . 3  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  e. 
_V )
15 simpl 457 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  ph )
161, 2, 3, 4, 5, 6, 7, 8, 9sibfima 28153 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) )
17 sitgclg.d . . . . . . . . . . 11  |-  D  =  ( ( dist `  G
)  |`  ( ( Base `  G )  X.  ( Base `  G ) ) )
18 sitgclg.g . . . . . . . . . . . . 13  |-  G  =  (Scalar `  W )
1918fveq2i 5859 . . . . . . . . . . . 12  |-  ( dist `  G )  =  (
dist `  (Scalar `  W
) )
2018fveq2i 5859 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  (Scalar `  W
) )
2120, 20xpeq12i 5011 . . . . . . . . . . . 12  |-  ( (
Base `  G )  X.  ( Base `  G
) )  =  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) )
2219, 21reseq12i 5261 . . . . . . . . . . 11  |-  ( (
dist `  G )  |`  ( ( Base `  G
)  X.  ( Base `  G ) ) )  =  ( ( dist `  (Scalar `  W )
)  |`  ( ( Base `  (Scalar `  W )
)  X.  ( Base `  (Scalar `  W )
) ) )
2317, 22eqtri 2472 . . . . . . . . . 10  |-  D  =  ( ( dist `  (Scalar `  W ) )  |`  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) ) )
24 eqid 2443 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
25 eqid 2443 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2618fveq2i 5859 . . . . . . . . . 10  |-  ( TopOpen `  G )  =  (
TopOpen `  (Scalar `  W
) )
2718fveq2i 5859 . . . . . . . . . 10  |-  ( ZMod
`  G )  =  ( ZMod `  (Scalar `  W ) )
28 sitgclg.3 . . . . . . . . . . . . 13  |-  ( ph  ->  (Scalar `  W )  e. ℝExt  )
2918, 28syl5eqel 2535 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. ℝExt  )
30 rrextdrg 27856 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e.  DivRing )
3129, 30syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  DivRing )
3218, 31syl5eqelr 2536 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e.  DivRing )
33 rrextnrg 27855 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e. NrmRing )
3429, 33syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  e. NrmRing )
3518, 34syl5eqelr 2536 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e. NrmRing )
36 eqid 2443 . . . . . . . . . . . 12  |-  ( ZMod
`  G )  =  ( ZMod `  G
)
3736rrextnlm 27857 . . . . . . . . . . 11  |-  ( G  e. ℝExt  ->  ( ZMod `  G )  e. NrmMod )
3829, 37syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ZMod `  G
)  e. NrmMod )
3918fveq2i 5859 . . . . . . . . . . 11  |-  (chr `  G )  =  (chr
`  (Scalar `  W )
)
40 rrextchr 27858 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  (chr `  G
)  =  0 )
4129, 40syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (chr `  G )  =  0 )
4239, 41syl5eqr 2498 . . . . . . . . . 10  |-  ( ph  ->  (chr `  (Scalar `  W
) )  =  0 )
43 rrextcusp 27859 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e. CUnifSp )
4429, 43syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  e. CUnifSp )
4518, 44syl5eqelr 2536 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e. CUnifSp )
4618fveq2i 5859 . . . . . . . . . . 11  |-  (UnifSt `  G )  =  (UnifSt `  (Scalar `  W )
)
47 eqid 2443 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
4847, 17rrextust 27862 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  (UnifSt `  G )  =  (metUnif `  D )
)
4929, 48syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
)
5046, 49syl5eqr 2498 . . . . . . . . . 10  |-  ( ph  ->  (UnifSt `  (Scalar `  W
) )  =  (metUnif `  D ) )
5123, 24, 25, 26, 27, 32, 35, 38, 42, 45, 50rrhf 27852 . . . . . . . . 9  |-  ( ph  ->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
526feq1i 5713 . . . . . . . . 9  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  <->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
5351, 52sylibr 212 . . . . . . . 8  |-  ( ph  ->  H : RR --> ( Base `  (Scalar `  W )
) )
54 ffun 5723 . . . . . . . 8  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  Fun  H )
5553, 54syl 16 . . . . . . 7  |-  ( ph  ->  Fun  H )
56 rge0ssre 11637 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
57 fdm 5725 . . . . . . . . 9  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  dom  H  =  RR )
5853, 57syl 16 . . . . . . . 8  |-  ( ph  ->  dom  H  =  RR )
5956, 58syl5sseqr 3538 . . . . . . 7  |-  ( ph  ->  ( 0 [,) +oo )  C_  dom  H )
60 funfvima2 6133 . . . . . . 7  |-  ( ( Fun  H  /\  (
0 [,) +oo )  C_ 
dom  H )  -> 
( ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
6155, 59, 60syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
6215, 16, 61sylc 60 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( H `  ( M `  ( `' F " { x }
) ) )  e.  ( H " (
0 [,) +oo )
) )
63 dmmeas 28045 . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . 15  |-  ( TopOpen `  W )  e.  _V
662, 65eqeltri 2527 . . . . . . . . . . . . . 14  |-  J  e. 
_V
6766a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  _V )
6867sgsiga 28015 . . . . . . . . . . . 12  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
693, 68syl5eqel 2535 . . . . . . . . . . 11  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
701, 2, 3, 4, 5, 6, 7, 8, 9sibfmbl 28150 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
7164, 69, 70mbfmf 28099 . . . . . . . . . 10  |-  ( ph  ->  F : U. dom  M --> U. S )
72 frn 5727 . . . . . . . . . 10  |-  ( F : U. dom  M --> U. S  ->  ran  F  C_ 
U. S )
7371, 72syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  U. S
)
743unieqi 4243 . . . . . . . . . . 11  |-  U. S  =  U. (sigaGen `  J
)
75 unisg 28016 . . . . . . . . . . . 12  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
7666, 75mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
7774, 76syl5eq 2496 . . . . . . . . . 10  |-  ( ph  ->  U. S  =  U. J )
78 sitgclg.1 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  TopSp )
791, 2tpsuni 19312 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
8078, 79syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
8177, 80eqtr4d 2487 . . . . . . . . 9  |-  ( ph  ->  U. S  =  B )
8273, 81sseqtrd 3525 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  B
)
8382ssdifd 3625 . . . . . . 7  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  C_  ( B  \  {  .0.  } ) )
8483sselda 3489 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  ( B  \  {  .0.  } ) )
8584eldifad 3473 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  B )
86 simp2 998 . . . . . 6  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) )  /\  x  e.  B )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H " ( 0 [,) +oo ) ) )
87 eleq1 2515 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  e.  ( H " (
0 [,) +oo )
)  <->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
88873anbi2d 1305 . . . . . . . 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 6288 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
9089eleq1d 2512 . . . . . . . 8  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( ( m 
.x.  x )  e.  B  <->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B ) )
9188, 90imbi12d 320 . . . . . . 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.4 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( H " ( 0 [,) +oo ) )  /\  x  e.  B
)  ->  ( m  .x.  x )  e.  B
)
9391, 92vtoclg 3153 . . . . . 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 36 . . . . 5  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) )  /\  x  e.  B )  ->  (
( H `  ( M `  ( `' F " { x }
) ) )  .x.  x )  e.  B
)
9515, 62, 85, 94syl3anc 1229 . . . 4  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B )
96 eqid 2443 . . . 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 6040 . . 3  |-  ( ph  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) : ( ran  F  \  {  .0.  } ) --> B )
98 mptexg 6127 . . . . . 6  |-  ( ( ran  F  \  {  .0.  } )  e.  _V  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) )  e.  _V )
9914, 98syl 16 . . . . 5  |-  ( ph  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) )  e.  _V )
100 fvex 5866 . . . . . 6  |-  ( 0g
`  W )  e. 
_V
1014, 100eqeltri 2527 . . . . 5  |-  .0.  e.  _V
102 suppimacnv 6914 . . . . 5  |-  ( ( ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) )  e.  _V  /\  .0.  e.  _V )  ->  ( ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) ) supp  .0.  )  =  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) ) )
10399, 101, 102sylancl 662 . . . 4  |-  ( ph  ->  ( ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) ) supp  .0.  )  =  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) ) )
1041, 2, 3, 4, 5, 6, 7, 8, 9sibfrn 28152 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
105 cnvimass 5347 . . . . . . 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 ) )
10696dmmptss 5493 . . . . . . 7  |-  dom  (
x  e.  ( ran 
F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) 
C_  ( ran  F  \  {  .0.  } )
107105, 106sstri 3498 . . . . . 6  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ( ran  F  \  {  .0.  } )
108 difss 3616 . . . . . 6  |-  ( ran 
F  \  {  .0.  } )  C_  ran  F
109107, 108sstri 3498 . . . . 5  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ran  F
110 ssfi 7742 . . . . 5  |-  ( ( 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 )
111104, 109, 110sylancl 662 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
112103, 111eqeltrd 2531 . . 3  |-  ( ph  ->  ( ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) ) supp  .0.  )  e.  Fin )
1131, 4, 11, 14, 97, 112gsumcl2 16796 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) )  e.  B )
11410, 113eqeltrd 2531 1  |-  ( ph  ->  ( ( Wsitg M
) `  F )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    \ cdif 3458    C_ wss 3461   {csn 4014   U.cuni 4234    |-> cmpt 4495    X. cxp 4987   `'ccnv 4988   dom cdm 4989   ran crn 4990    |` cres 4991   "cima 4992   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   supp csupp 6903   Fincfn 7518   RRcr 9494   0cc0 9495   +oocpnf 9628   (,)cioo 11538   [,)cico 11540   Basecbs 14509  Scalarcsca 14577   .scvsca 14578   distcds 14583   TopOpenctopn 14696   topGenctg 14712   0gc0g 14714    gsumg cgsu 14715  CMndccmn 16672   DivRingcdr 17270  metUnifcmetu 18284   ZModczlm 18411  chrcchr 18412   TopSpctps 19270  UnifStcuss 20629  CUnifSpccusp 20673  NrmRingcnrg 20973  NrmModcnlm 20974  RRHomcrrh 27847   ℝExt crrext 27848  sigAlgebracsiga 27980  sigaGencsigagen 28011  measurescmeas 28039  sitgcsitg 28144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-dvds 13864  df-gcd 14022  df-numer 14145  df-denom 14146  df-gz 14325  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-subg 16072  df-ghm 16139  df-cntz 16229  df-od 16427  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-dvr 17206  df-rnghom 17238  df-drng 17272  df-subrg 17301  df-abv 17340  df-lmod 17388  df-nzr 17780  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-metu 18293  df-cnfld 18295  df-zring 18363  df-zrh 18414  df-zlm 18415  df-chr 18416  df-refld 18514  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-cn 19601  df-cnp 19602  df-haus 19689  df-reg 19690  df-cmp 19760  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-fcls 20315  df-cnext 20433  df-ust 20576  df-utop 20607  df-uss 20632  df-usp 20633  df-ucn 20652  df-cfilu 20663  df-cusp 20674  df-xms 20696  df-ms 20697  df-tms 20698  df-nm 20976  df-ngp 20977  df-nrg 20979  df-nlm 20980  df-cncf 21255  df-cfil 21567  df-cmet 21569  df-cms 21647  df-qqh 27827  df-rrh 27849  df-rrext 27853  df-esum 27914  df-siga 27981  df-sigagen 28012  df-meas 28040  df-mbfm 28095  df-sitg 28145
This theorem is referenced by:  sitgclbn  28158
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