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Theorem sitgclg 26728
Description: Closure of the Bochner integral on simple functions, generic version. See sitgclbn 26729 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  ->  (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
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 26727 . 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 6510 . . . 4  |-  ( F  e.  dom  ( Wsitg M )  ->  ran  F  e.  _V )
13 difexg 4440 . . . 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 26724 . . . . . 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 5694 . . . . . . . . . . . . 13  |-  ( dist `  G )  =  (
dist `  (Scalar `  W
) )
2018fveq2i 5694 . . . . . . . . . . . . . 14  |-  ( Base `  G )  =  (
Base `  (Scalar `  W
) )
2120, 20xpeq12i 4862 . . . . . . . . . . . . 13  |-  ( (
Base `  G )  X.  ( Base `  G
) )  =  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) )
2219, 21reseq12i 5108 . . . . . . . . . . . 12  |-  ( (
dist `  G )  |`  ( ( Base `  G
)  X.  ( Base `  G ) ) )  =  ( ( dist `  (Scalar `  W )
)  |`  ( ( Base `  (Scalar `  W )
)  X.  ( Base `  (Scalar `  W )
) ) )
2317, 22eqtri 2463 . . . . . . . . . . 11  |-  D  =  ( ( dist `  (Scalar `  W ) )  |`  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) ) )
24 eqid 2443 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
25 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2618fveq2i 5694 . . . . . . . . . . 11  |-  ( TopOpen `  G )  =  (
TopOpen `  (Scalar `  W
) )
2718fveq2i 5694 . . . . . . . . . . 11  |-  ( ZMod
`  G )  =  ( ZMod `  (Scalar `  W ) )
28 sitgclg.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  (Scalar `  W )  e. ℝExt  )
2918, 28syl5eqel 2527 . . . . . . . . . . . . 13  |-  ( ph  ->  G  e. ℝExt  )
30 rrextdrg 26431 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  G  e.  DivRing )
3129, 30syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  DivRing )
3218, 31syl5eqelr 2528 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e.  DivRing )
33 rrextnrg 26430 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  G  e. NrmRing )
3429, 33syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. NrmRing )
3518, 34syl5eqelr 2528 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e. NrmRing )
36 eqid 2443 . . . . . . . . . . . . 13  |-  ( ZMod
`  G )  =  ( ZMod `  G
)
3736rrextnlm 26432 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  ( ZMod `  G )  e. NrmMod )
3829, 37syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ZMod `  G
)  e. NrmMod )
3918fveq2i 5694 . . . . . . . . . . . 12  |-  (chr `  G )  =  (chr
`  (Scalar `  W )
)
40 rrextchr 26433 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  (chr `  G
)  =  0 )
4129, 40syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  (chr `  G )  =  0 )
4239, 41syl5eqr 2489 . . . . . . . . . . 11  |-  ( ph  ->  (chr `  (Scalar `  W
) )  =  0 )
43 rrextcusp 26434 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  G  e. CUnifSp )
4429, 43syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. CUnifSp )
4518, 44syl5eqelr 2528 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e. CUnifSp )
4618fveq2i 5694 . . . . . . . . . . . 12  |-  (UnifSt `  G )  =  (UnifSt `  (Scalar `  W )
)
47 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Base `  G )  =  (
Base `  G )
4847, 17rrextust 26437 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  (UnifSt `  G )  =  (metUnif `  D )
)
4929, 48syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
)
5046, 49syl5eqr 2489 . . . . . . . . . . 11  |-  ( ph  ->  (UnifSt `  (Scalar `  W
) )  =  (metUnif `  D ) )
5123, 24, 25, 26, 27, 32, 35, 38, 42, 45, 50rrhf 26427 . . . . . . . . . 10  |-  ( ph  ->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
526feq1i 5551 . . . . . . . . . 10  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  <->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
5351, 52sylibr 212 . . . . . . . . 9  |-  ( ph  ->  H : RR --> ( Base `  (Scalar `  W )
) )
54 ffun 5561 . . . . . . . . 9  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  Fun  H )
5553, 54syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  H )
56 0re 9386 . . . . . . . . . . 11  |-  0  e.  RR
57 pnfxr 11092 . . . . . . . . . . 11  |- +oo  e.  RR*
58 icossre 11376 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
5956, 57, 58mp2an 672 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
6059a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( 0 [,) +oo )  C_  RR )
61 fdm 5563 . . . . . . . . . 10  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  dom  H  =  RR )
6253, 61syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  H  =  RR )
6360, 62sseqtr4d 3393 . . . . . . . 8  |-  ( ph  ->  ( 0 [,) +oo )  C_  dom  H )
64 funfvima2 5953 . . . . . . . 8  |-  ( ( Fun  H  /\  (
0 [,) +oo )  C_ 
dom  H )  -> 
( ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
6555, 63, 64syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
6665imp 429 . . . . . 6  |-  ( (
ph  /\  ( M `  ( `' F " { x } ) )  e.  ( 0 [,) +oo ) )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) )
6715, 16, 66syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( H `  ( M `  ( `' F " { x }
) ) )  e.  ( H " (
0 [,) +oo )
) )
68 isrnmeas 26614 . . . . . . . . . . . . 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 ) ) ) ) )
6968simpld 459 . . . . . . . . . . . 12  |-  ( M  e.  U. ran measures  ->  dom  M  e.  U. ran sigAlgebra )
708, 69syl 16 . . . . . . . . . . 11  |-  ( ph  ->  dom  M  e.  U. ran sigAlgebra )
71 fvex 5701 . . . . . . . . . . . . . . 15  |-  ( TopOpen `  W )  e.  _V
722, 71eqeltri 2513 . . . . . . . . . . . . . 14  |-  J  e. 
_V
7372a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  _V )
7473sgsiga 26585 . . . . . . . . . . . 12  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
753, 74syl5eqel 2527 . . . . . . . . . . 11  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
761, 2, 3, 4, 5, 6, 7, 8, 9sibfmbl 26721 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
7770, 75, 76mbfmf 26670 . . . . . . . . . 10  |-  ( ph  ->  F : U. dom  M --> U. S )
78 frn 5565 . . . . . . . . . 10  |-  ( F : U. dom  M --> U. S  ->  ran  F  C_ 
U. S )
7977, 78syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  U. S
)
803unieqi 4100 . . . . . . . . . . 11  |-  U. S  =  U. (sigaGen `  J
)
81 unisg 26586 . . . . . . . . . . . 12  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
8272, 81mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
8380, 82syl5eq 2487 . . . . . . . . . 10  |-  ( ph  ->  U. S  =  U. J )
84 sitgclg.1 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  TopSp )
851, 2tpsuni 18543 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
8684, 85syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
8783, 86eqtr4d 2478 . . . . . . . . 9  |-  ( ph  ->  U. S  =  B )
8879, 87sseqtrd 3392 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  B
)
8988ssdifd 3492 . . . . . . 7  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  C_  ( B  \  {  .0.  } ) )
9089sselda 3356 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  ( B  \  {  .0.  } ) )
9190eldifad 3340 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  B )
92 simp2 989 . . . . . 6  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) )  /\  x  e.  B )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H " ( 0 [,) +oo ) ) )
93 eleq1 2503 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  e.  ( H " (
0 [,) +oo )
)  <->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
94933anbi2d 1294 . . . . . . . 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 ) ) )
95 oveq1 6098 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
9695eleq1d 2509 . . . . . . . 8  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( ( m 
.x.  x )  e.  B  <->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B ) )
9794, 96imbi12d 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
) ) )
98 sitgclg.4 . . . . . . 7  |-  ( (
ph  /\  m  e.  ( H " ( 0 [,) +oo ) )  /\  x  e.  B
)  ->  ( m  .x.  x )  e.  B
)
9997, 98vtoclg 3030 . . . . . 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
) )
10092, 99mpcom 36 . . . . 5  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) )  /\  x  e.  B )  ->  (
( H `  ( M `  ( `' F " { x }
) ) )  .x.  x )  e.  B
)
10115, 67, 91, 100syl3anc 1218 . . . 4  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B )
102 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 ) )
103101, 102fmptd 5867 . . 3  |-  ( ph  ->  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) : ( ran  F  \  {  .0.  } ) --> B )
1041, 2, 3, 4, 5, 6, 7, 8, 9sibfrn 26723 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
105 cnvimass 5189 . . . . . . 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 ) )
106102dmmptss 5334 . . . . . . 7  |-  dom  (
x  e.  ( ran 
F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) 
C_  ( ran  F  \  {  .0.  } )
107105, 106sstri 3365 . . . . . 6  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ( ran  F  \  {  .0.  } )
108 difss 3483 . . . . . 6  |-  ( ran 
F  \  {  .0.  } )  C_  ran  F
109107, 108sstri 3365 . . . . 5  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ran  F
110109a1i 11 . . . 4  |-  ( ph  ->  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) ) 
C_  ran  F )
111 ssfi 7533 . . . 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 )
112104, 110, 111syl2anc 661 . . 3  |-  ( ph  ->  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
1131, 4, 11, 14, 103, 112gsumclOLD 16400 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) )  e.  B )
11410, 113eqeltrd 2517 1  |-  ( ph  ->  ( ( Wsitg M
) `  F )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    \ cdif 3325    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   {csn 3877   U.cuni 4091  Disj wdisj 4262   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   `'ccnv 4839   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843   Fun wfun 5412   -->wf 5414   ` cfv 5418  (class class class)co 6091   omcom 6476    ~<_ cdom 7308   Fincfn 7310   RRcr 9281   0cc0 9282   +oocpnf 9415   RR*cxr 9417   (,)cioo 11300   [,)cico 11302   [,]cicc 11303   Basecbs 14174  Scalarcsca 14241   .scvsca 14242   distcds 14247   TopOpenctopn 14360   topGenctg 14376   0gc0g 14378    gsumg cgsu 14379  CMndccmn 16277   DivRingcdr 16832  metUnifcmetu 17808   ZModczlm 17932  chrcchr 17933   TopSpctps 18501  UnifStcuss 19828  CUnifSpccusp 19872  NrmRingcnrg 20172  NrmModcnlm 20173  RRHomcrrh 26422   ℝExt crrext 26423  Σ*cesum 26483  sigAlgebracsiga 26550  sigaGencsigagen 26581  measurescmeas 26609  sitgcsitg 26715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-numer 13813  df-denom 13814  df-gz 13991  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-subg 15678  df-ghm 15745  df-cntz 15835  df-od 16032  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-rnghom 16806  df-drng 16834  df-subrg 16863  df-abv 16902  df-lmod 16950  df-nzr 17340  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-metu 17817  df-cnfld 17819  df-zring 17884  df-zrh 17935  df-zlm 17936  df-chr 17937  df-refld 18035  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-cn 18831  df-cnp 18832  df-haus 18919  df-reg 18920  df-cmp 18990  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-fcls 19514  df-cnext 19632  df-ust 19775  df-utop 19806  df-uss 19831  df-usp 19832  df-ucn 19851  df-cfilu 19862  df-cusp 19873  df-xms 19895  df-ms 19896  df-tms 19897  df-nm 20175  df-ngp 20176  df-nrg 20178  df-nlm 20179  df-cncf 20454  df-cfil 20766  df-cmet 20768  df-cms 20846  df-qqh 26402  df-rrh 26424  df-rrext 26428  df-esum 26484  df-siga 26551  df-sigagen 26582  df-meas 26610  df-mbfm 26666  df-sitg 26716
This theorem is referenced by:  sitgclbn  26729
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