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Theorem sitgclg 29171
Description: Closure of the Bochner integral on simple functions, generic version. See sitgclbn 29172 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 29170 . 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 6736 . . . 4  |-  ( F  e.  dom  ( Wsitg M )  ->  ran  F  e.  _V )
13 difexg 4569 . . . 4  |-  ( ran 
F  e.  _V  ->  ( ran  F  \  {  .0.  } )  e.  _V )
149, 12, 133syl 18 . . 3  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  e. 
_V )
15 simpl 458 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  ph )
161, 2, 3, 4, 5, 6, 7, 8, 9sibfima 29167 . . . . . 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 5881 . . . . . . . . . . . 12  |-  ( dist `  G )  =  (
dist `  (Scalar `  W
) )
2018fveq2i 5881 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  (Scalar `  W
) )
2120, 20xpeq12i 4872 . . . . . . . . . . . 12  |-  ( (
Base `  G )  X.  ( Base `  G
) )  =  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) )
2219, 21reseq12i 5119 . . . . . . . . . . 11  |-  ( (
dist `  G )  |`  ( ( Base `  G
)  X.  ( Base `  G ) ) )  =  ( ( dist `  (Scalar `  W )
)  |`  ( ( Base `  (Scalar `  W )
)  X.  ( Base `  (Scalar `  W )
) ) )
2317, 22eqtri 2451 . . . . . . . . . 10  |-  D  =  ( ( dist `  (Scalar `  W ) )  |`  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) ) )
24 eqid 2422 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
25 eqid 2422 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2618fveq2i 5881 . . . . . . . . . 10  |-  ( TopOpen `  G )  =  (
TopOpen `  (Scalar `  W
) )
2718fveq2i 5881 . . . . . . . . . 10  |-  ( ZMod
`  G )  =  ( ZMod `  (Scalar `  W ) )
28 sitgclg.3 . . . . . . . . . . . . 13  |-  ( ph  ->  (Scalar `  W )  e. ℝExt  )
2918, 28syl5eqel 2514 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. ℝExt  )
30 rrextdrg 28802 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e.  DivRing )
3129, 30syl 17 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  DivRing )
3218, 31syl5eqelr 2515 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e.  DivRing )
33 rrextnrg 28801 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e. NrmRing )
3429, 33syl 17 . . . . . . . . . . 11  |-  ( ph  ->  G  e. NrmRing )
3518, 34syl5eqelr 2515 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e. NrmRing )
36 eqid 2422 . . . . . . . . . . . 12  |-  ( ZMod
`  G )  =  ( ZMod `  G
)
3736rrextnlm 28803 . . . . . . . . . . 11  |-  ( G  e. ℝExt  ->  ( ZMod `  G )  e. NrmMod )
3829, 37syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( ZMod `  G
)  e. NrmMod )
3918fveq2i 5881 . . . . . . . . . . 11  |-  (chr `  G )  =  (chr
`  (Scalar `  W )
)
40 rrextchr 28804 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  (chr `  G
)  =  0 )
4129, 40syl 17 . . . . . . . . . . 11  |-  ( ph  ->  (chr `  G )  =  0 )
4239, 41syl5eqr 2477 . . . . . . . . . 10  |-  ( ph  ->  (chr `  (Scalar `  W
) )  =  0 )
43 rrextcusp 28805 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e. CUnifSp )
4429, 43syl 17 . . . . . . . . . . 11  |-  ( ph  ->  G  e. CUnifSp )
4518, 44syl5eqelr 2515 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e. CUnifSp )
4618fveq2i 5881 . . . . . . . . . . 11  |-  (UnifSt `  G )  =  (UnifSt `  (Scalar `  W )
)
47 eqid 2422 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
4847, 17rrextust 28808 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  (UnifSt `  G )  =  (metUnif `  D )
)
4929, 48syl 17 . . . . . . . . . . 11  |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
)
5046, 49syl5eqr 2477 . . . . . . . . . 10  |-  ( ph  ->  (UnifSt `  (Scalar `  W
) )  =  (metUnif `  D ) )
5123, 24, 25, 26, 27, 32, 35, 38, 42, 45, 50rrhf 28798 . . . . . . . . 9  |-  ( ph  ->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
526feq1i 5735 . . . . . . . . 9  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  <->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
5351, 52sylibr 215 . . . . . . . 8  |-  ( ph  ->  H : RR --> ( Base `  (Scalar `  W )
) )
54 ffun 5745 . . . . . . . 8  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  Fun  H )
5553, 54syl 17 . . . . . . 7  |-  ( ph  ->  Fun  H )
56 rge0ssre 11741 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
57 fdm 5747 . . . . . . . . 9  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  dom  H  =  RR )
5853, 57syl 17 . . . . . . . 8  |-  ( ph  ->  dom  H  =  RR )
5956, 58syl5sseqr 3513 . . . . . . 7  |-  ( ph  ->  ( 0 [,) +oo )  C_  dom  H )
60 funfvima2 6153 . . . . . . 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 665 . . . . . 6  |-  ( ph  ->  ( ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
6215, 16, 61sylc 62 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( H `  ( M `  ( `' F " { x }
) ) )  e.  ( H " (
0 [,) +oo )
) )
63 dmmeas 29019 . . . . . . . . . . . 12  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
648, 63syl 17 . . . . . . . . . . 11  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
65 fvex 5888 . . . . . . . . . . . . . . 15  |-  ( TopOpen `  W )  e.  _V
662, 65eqeltri 2506 . . . . . . . . . . . . . 14  |-  J  e. 
_V
6766a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  _V )
6867sgsiga 28960 . . . . . . . . . . . 12  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
693, 68syl5eqel 2514 . . . . . . . . . . 11  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
701, 2, 3, 4, 5, 6, 7, 8, 9sibfmbl 29164 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
7164, 69, 70mbfmf 29073 . . . . . . . . . 10  |-  ( ph  ->  F : U. dom  M --> U. S )
72 frn 5749 . . . . . . . . . 10  |-  ( F : U. dom  M --> U. S  ->  ran  F  C_ 
U. S )
7371, 72syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  U. S
)
743unieqi 4225 . . . . . . . . . . 11  |-  U. S  =  U. (sigaGen `  J
)
75 unisg 28961 . . . . . . . . . . . 12  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
7666, 75mp1i 13 . . . . . . . . . . 11  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
7774, 76syl5eq 2475 . . . . . . . . . 10  |-  ( ph  ->  U. S  =  U. J )
78 sitgclg.1 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  TopSp )
791, 2tpsuni 19940 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
8078, 79syl 17 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
8177, 80eqtr4d 2466 . . . . . . . . 9  |-  ( ph  ->  U. S  =  B )
8273, 81sseqtrd 3500 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  B
)
8382ssdifd 3601 . . . . . . 7  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  C_  ( B  \  {  .0.  } ) )
8483sselda 3464 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  ( B  \  {  .0.  } ) )
8584eldifad 3448 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  B )
86 simp2 1006 . . . . . 6  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) )  /\  x  e.  B )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H " ( 0 [,) +oo ) ) )
87 eleq1 2494 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  e.  ( H " (
0 [,) +oo )
)  <->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
88873anbi2d 1340 . . . . . . . 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 6309 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
9089eleq1d 2491 . . . . . . . 8  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( ( m 
.x.  x )  e.  B  <->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B ) )
9188, 90imbi12d 321 . . . . . . 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 3139 . . . . . 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 37 . . . . 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 1264 . . . 4  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B )
96 eqid 2422 . . . 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 6058 . . 3  |-  ( ph  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) : ( ran  F  \  {  .0.  } ) --> B )
98 mptexg 6147 . . . . . 6  |-  ( ( ran  F  \  {  .0.  } )  e.  _V  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) )  e.  _V )
9914, 98syl 17 . . . . 5  |-  ( ph  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) )  e.  _V )
100 fvex 5888 . . . . . 6  |-  ( 0g
`  W )  e. 
_V
1014, 100eqeltri 2506 . . . . 5  |-  .0.  e.  _V
102 suppimacnv 6933 . . . . 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 666 . . . 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 29166 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
105 cnvimass 5204 . . . . . . 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 5347 . . . . . . 7  |-  dom  (
x  e.  ( ran 
F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) 
C_  ( ran  F  \  {  .0.  } )
107105, 106sstri 3473 . . . . . 6  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ( ran  F  \  {  .0.  } )
108 difss 3592 . . . . . 6  |-  ( ran 
F  \  {  .0.  } )  C_  ran  F
109107, 108sstri 3473 . . . . 5  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ran  F
110 ssfi 7795 . . . . 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 666 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
112103, 111eqeltrd 2510 . . 3  |-  ( ph  ->  ( ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) ) supp  .0.  )  e.  Fin )
1131, 4, 11, 14, 97, 112gsumcl2 17536 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) )  e.  B )
11410, 113eqeltrd 2510 1  |-  ( ph  ->  ( ( Wsitg M
) `  F )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   _Vcvv 3081    \ cdif 3433    C_ wss 3436   {csn 3996   U.cuni 4216    |-> cmpt 4479    X. cxp 4848   `'ccnv 4849   dom cdm 4850   ran crn 4851    |` cres 4852   "cima 4853   Fun wfun 5592   -->wf 5594   ` cfv 5598  (class class class)co 6302   supp csupp 6922   Fincfn 7574   RRcr 9539   0cc0 9540   +oocpnf 9673   (,)cioo 11636   [,)cico 11638   Basecbs 15109  Scalarcsca 15181   .scvsca 15182   distcds 15187   TopOpenctopn 15308   topGenctg 15324   0gc0g 15326    gsumg cgsu 15327  CMndccmn 17418   DivRingcdr 17963  metUnifcmetu 18949   ZModczlm 19059  chrcchr 19060   TopSpctps 19906  UnifStcuss 21255  CUnifSpccusp 21299  NrmRingcnrg 21581  NrmModcnlm 21582  RRHomcrrh 28793   ℝExt crrext 28794  sigAlgebracsiga 28925  sigaGencsigagen 28956  measurescmeas 29013  sitgcsitg 29158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-dvds 14294  df-gcd 14457  df-numer 14672  df-denom 14673  df-gz 14862  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-mhm 16570  df-submnd 16571  df-grp 16661  df-minusg 16662  df-sbg 16663  df-mulg 16664  df-subg 16802  df-ghm 16869  df-cntz 16959  df-od 17160  df-cmn 17420  df-abl 17421  df-mgp 17712  df-ur 17724  df-ring 17770  df-cring 17771  df-oppr 17839  df-dvdsr 17857  df-unit 17858  df-invr 17888  df-dvr 17899  df-rnghom 17931  df-drng 17965  df-subrg 17994  df-abv 18033  df-lmod 18081  df-nzr 18470  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-fbas 18955  df-fg 18956  df-metu 18957  df-cnfld 18959  df-zring 19027  df-zrh 19062  df-zlm 19063  df-chr 19064  df-refld 19160  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-nei 20101  df-cn 20230  df-cnp 20231  df-haus 20318  df-reg 20319  df-cmp 20389  df-tx 20564  df-hmeo 20757  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-fcls 20943  df-cnext 21062  df-ust 21202  df-utop 21233  df-uss 21258  df-usp 21259  df-ucn 21278  df-cfilu 21289  df-cusp 21300  df-xms 21322  df-ms 21323  df-tms 21324  df-nm 21584  df-ngp 21585  df-nrg 21587  df-nlm 21588  df-cncf 21897  df-cfil 22212  df-cmet 22214  df-cms 22290  df-qqh 28773  df-rrh 28795  df-rrext 28799  df-esum 28845  df-siga 28926  df-sigagen 28957  df-meas 29014  df-mbfm 29069  df-sitg 29159
This theorem is referenced by:  sitgclbn  29172  sitmcl  29180
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