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Theorem sitgclg 27952
Description: Closure of the Bochner integral on simple functions, generic version. See sitgclbn 27953 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 27951 . 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 6716 . . . 4  |-  ( F  e.  dom  ( Wsitg M )  ->  ran  F  e.  _V )
13 difexg 4595 . . . 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 27948 . . . . . 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 5869 . . . . . . . . . . . 12  |-  ( dist `  G )  =  (
dist `  (Scalar `  W
) )
2018fveq2i 5869 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  (Scalar `  W
) )
2120, 20xpeq12i 5021 . . . . . . . . . . . 12  |-  ( (
Base `  G )  X.  ( Base `  G
) )  =  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) )
2219, 21reseq12i 5271 . . . . . . . . . . 11  |-  ( (
dist `  G )  |`  ( ( Base `  G
)  X.  ( Base `  G ) ) )  =  ( ( dist `  (Scalar `  W )
)  |`  ( ( Base `  (Scalar `  W )
)  X.  ( Base `  (Scalar `  W )
) ) )
2317, 22eqtri 2496 . . . . . . . . . 10  |-  D  =  ( ( dist `  (Scalar `  W ) )  |`  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) ) )
24 eqid 2467 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
25 eqid 2467 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2618fveq2i 5869 . . . . . . . . . 10  |-  ( TopOpen `  G )  =  (
TopOpen `  (Scalar `  W
) )
2718fveq2i 5869 . . . . . . . . . 10  |-  ( ZMod
`  G )  =  ( ZMod `  (Scalar `  W ) )
28 sitgclg.3 . . . . . . . . . . . . 13  |-  ( ph  ->  (Scalar `  W )  e. ℝExt  )
2918, 28syl5eqel 2559 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. ℝExt  )
30 rrextdrg 27647 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e.  DivRing )
3129, 30syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  DivRing )
3218, 31syl5eqelr 2560 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e.  DivRing )
33 rrextnrg 27646 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e. NrmRing )
3429, 33syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  e. NrmRing )
3518, 34syl5eqelr 2560 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e. NrmRing )
36 eqid 2467 . . . . . . . . . . . 12  |-  ( ZMod
`  G )  =  ( ZMod `  G
)
3736rrextnlm 27648 . . . . . . . . . . 11  |-  ( G  e. ℝExt  ->  ( ZMod `  G )  e. NrmMod )
3829, 37syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ZMod `  G
)  e. NrmMod )
3918fveq2i 5869 . . . . . . . . . . 11  |-  (chr `  G )  =  (chr
`  (Scalar `  W )
)
40 rrextchr 27649 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  (chr `  G
)  =  0 )
4129, 40syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (chr `  G )  =  0 )
4239, 41syl5eqr 2522 . . . . . . . . . 10  |-  ( ph  ->  (chr `  (Scalar `  W
) )  =  0 )
43 rrextcusp 27650 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  G  e. CUnifSp )
4429, 43syl 16 . . . . . . . . . . 11  |-  ( ph  ->  G  e. CUnifSp )
4518, 44syl5eqelr 2560 . . . . . . . . . 10  |-  ( ph  ->  (Scalar `  W )  e. CUnifSp )
4618fveq2i 5869 . . . . . . . . . . 11  |-  (UnifSt `  G )  =  (UnifSt `  (Scalar `  W )
)
47 eqid 2467 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
4847, 17rrextust 27653 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  (UnifSt `  G )  =  (metUnif `  D )
)
4929, 48syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
)
5046, 49syl5eqr 2522 . . . . . . . . . 10  |-  ( ph  ->  (UnifSt `  (Scalar `  W
) )  =  (metUnif `  D ) )
5123, 24, 25, 26, 27, 32, 35, 38, 42, 45, 50rrhf 27643 . . . . . . . . 9  |-  ( ph  ->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
526feq1i 5723 . . . . . . . . 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 5733 . . . . . . . 8  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  Fun  H )
5553, 54syl 16 . . . . . . 7  |-  ( ph  ->  Fun  H )
56 rge0ssre 11628 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
57 fdm 5735 . . . . . . . . 9  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  dom  H  =  RR )
5853, 57syl 16 . . . . . . . 8  |-  ( ph  ->  dom  H  =  RR )
5956, 58syl5sseqr 3553 . . . . . . 7  |-  ( ph  ->  ( 0 [,) +oo )  C_  dom  H )
60 funfvima2 6136 . . . . . . 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 27840 . . . . . . . . . . . 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 5876 . . . . . . . . . . . . . . 15  |-  ( TopOpen `  W )  e.  _V
662, 65eqeltri 2551 . . . . . . . . . . . . . 14  |-  J  e. 
_V
6766a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  _V )
6867sgsiga 27810 . . . . . . . . . . . 12  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
693, 68syl5eqel 2559 . . . . . . . . . . 11  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
701, 2, 3, 4, 5, 6, 7, 8, 9sibfmbl 27945 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
7164, 69, 70mbfmf 27894 . . . . . . . . . 10  |-  ( ph  ->  F : U. dom  M --> U. S )
72 frn 5737 . . . . . . . . . 10  |-  ( F : U. dom  M --> U. S  ->  ran  F  C_ 
U. S )
7371, 72syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  U. S
)
743unieqi 4254 . . . . . . . . . . 11  |-  U. S  =  U. (sigaGen `  J
)
75 unisg 27811 . . . . . . . . . . . 12  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
7666, 75mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
7774, 76syl5eq 2520 . . . . . . . . . 10  |-  ( ph  ->  U. S  =  U. J )
78 sitgclg.1 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  TopSp )
791, 2tpsuni 19234 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
8078, 79syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
8177, 80eqtr4d 2511 . . . . . . . . 9  |-  ( ph  ->  U. S  =  B )
8273, 81sseqtrd 3540 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  B
)
8382ssdifd 3640 . . . . . . 7  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  C_  ( B  \  {  .0.  } ) )
8483sselda 3504 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  ( B  \  {  .0.  } ) )
8584eldifad 3488 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  B )
86 simp2 997 . . . . . 6  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) )  /\  x  e.  B )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H " ( 0 [,) +oo ) ) )
87 eleq1 2539 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  e.  ( H " (
0 [,) +oo )
)  <->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
88873anbi2d 1304 . . . . . . . 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 6291 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
9089eleq1d 2536 . . . . . . . 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 3171 . . . . . 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 1228 . . . 4  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B )
96 eqid 2467 . . . 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 6045 . . 3  |-  ( ph  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) : ( ran  F  \  {  .0.  } ) --> B )
98 mptexg 6130 . . . . . 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 5876 . . . . . 6  |-  ( 0g
`  W )  e. 
_V
1014, 100eqeltri 2551 . . . . 5  |-  .0.  e.  _V
102 suppimacnv 6912 . . . . 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 27947 . . . . 5  |-  ( ph  ->  ran  F  e.  Fin )
105 cnvimass 5357 . . . . . . 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 5503 . . . . . . 7  |-  dom  (
x  e.  ( ran 
F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) 
C_  ( ran  F  \  {  .0.  } )
107105, 106sstri 3513 . . . . . 6  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ( ran  F  \  {  .0.  } )
108 difss 3631 . . . . . 6  |-  ( ran 
F  \  {  .0.  } )  C_  ran  F
109107, 108sstri 3513 . . . . 5  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ran  F
110 ssfi 7740 . . . . 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 2555 . . 3  |-  ( ph  ->  ( ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) ) supp  .0.  )  e.  Fin )
1131, 4, 11, 14, 97, 112gsumcl2 16725 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) )  e.  B )
11410, 113eqeltrd 2555 1  |-  ( ph  ->  ( ( Wsitg M
) `  F )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473    C_ wss 3476   {csn 4027   U.cuni 4245    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6284   supp csupp 6901   Fincfn 7516   RRcr 9491   0cc0 9492   +oocpnf 9625   (,)cioo 11529   [,)cico 11531   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   distcds 14564   TopOpenctopn 14677   topGenctg 14693   0gc0g 14695    gsumg cgsu 14696  CMndccmn 16604   DivRingcdr 17196  metUnifcmetu 18209   ZModczlm 18333  chrcchr 18334   TopSpctps 19192  UnifStcuss 20519  CUnifSpccusp 20563  NrmRingcnrg 20863  NrmModcnlm 20864  RRHomcrrh 27638   ℝExt crrext 27639  sigAlgebracsiga 27775  sigaGencsigagen 27806  measurescmeas 27834  sitgcsitg 27939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-numer 14127  df-denom 14128  df-gz 14307  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-ghm 16070  df-cntz 16160  df-od 16359  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-rnghom 17165  df-drng 17198  df-subrg 17227  df-abv 17266  df-lmod 17314  df-nzr 17705  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-metu 18218  df-cnfld 18220  df-zring 18285  df-zrh 18336  df-zlm 18337  df-chr 18338  df-refld 18436  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-cn 19522  df-cnp 19523  df-haus 19610  df-reg 19611  df-cmp 19681  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-fcls 20205  df-cnext 20323  df-ust 20466  df-utop 20497  df-uss 20522  df-usp 20523  df-ucn 20542  df-cfilu 20553  df-cusp 20564  df-xms 20586  df-ms 20587  df-tms 20588  df-nm 20866  df-ngp 20867  df-nrg 20869  df-nlm 20870  df-cncf 21145  df-cfil 21457  df-cmet 21459  df-cms 21537  df-qqh 27618  df-rrh 27640  df-rrext 27644  df-esum 27709  df-siga 27776  df-sigagen 27807  df-meas 27835  df-mbfm 27890  df-sitg 27940
This theorem is referenced by:  sitgclbn  27953
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