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Theorem sitgclg 26658
Description: Closure of the Bochner integral on simple functions, generic version. See sitgclbn 26659 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 26657 . 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 6509 . . . 4  |-  ( F  e.  dom  ( Wsitg M )  ->  ran  F  e.  _V )
13 difexg 4437 . . . 4  |-  ( ran 
F  e.  _V  ->  ( ran  F  \  {  .0.  } )  e.  _V )
149, 12, 133syl 20 . . 3  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  e. 
_V )
15 simpl 454 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  ph )
161, 2, 3, 4, 5, 6, 7, 8, 9sibfima 26654 . . . . . 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 5691 . . . . . . . . . . . . 13  |-  ( dist `  G )  =  (
dist `  (Scalar `  W
) )
2018fveq2i 5691 . . . . . . . . . . . . . 14  |-  ( Base `  G )  =  (
Base `  (Scalar `  W
) )
2120, 20xpeq12i 4858 . . . . . . . . . . . . 13  |-  ( (
Base `  G )  X.  ( Base `  G
) )  =  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) )
2219, 21reseq12i 5104 . . . . . . . . . . . 12  |-  ( (
dist `  G )  |`  ( ( Base `  G
)  X.  ( Base `  G ) ) )  =  ( ( dist `  (Scalar `  W )
)  |`  ( ( Base `  (Scalar `  W )
)  X.  ( Base `  (Scalar `  W )
) ) )
2317, 22eqtri 2461 . . . . . . . . . . 11  |-  D  =  ( ( dist `  (Scalar `  W ) )  |`  ( ( Base `  (Scalar `  W ) )  X.  ( Base `  (Scalar `  W ) ) ) )
24 eqid 2441 . . . . . . . . . . 11  |-  ( topGen ` 
ran  (,) )  =  (
topGen `  ran  (,) )
25 eqid 2441 . . . . . . . . . . 11  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2618fveq2i 5691 . . . . . . . . . . 11  |-  ( TopOpen `  G )  =  (
TopOpen `  (Scalar `  W
) )
2718fveq2i 5691 . . . . . . . . . . 11  |-  ( ZMod
`  G )  =  ( ZMod `  (Scalar `  W ) )
28 sitgclg.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  (Scalar `  W )  e. ℝExt  )
2918, 28syl5eqel 2525 . . . . . . . . . . . . 13  |-  ( ph  ->  G  e. ℝExt  )
30 rrextdrg 26367 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  G  e.  DivRing )
3129, 30syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  DivRing )
3218, 31syl5eqelr 2526 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e.  DivRing )
33 rrextnrg 26366 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  G  e. NrmRing )
3429, 33syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. NrmRing )
3518, 34syl5eqelr 2526 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e. NrmRing )
36 eqid 2441 . . . . . . . . . . . . 13  |-  ( ZMod
`  G )  =  ( ZMod `  G
)
3736rrextnlm 26368 . . . . . . . . . . . 12  |-  ( G  e. ℝExt  ->  ( ZMod `  G )  e. NrmMod )
3829, 37syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ZMod `  G
)  e. NrmMod )
3918fveq2i 5691 . . . . . . . . . . . 12  |-  (chr `  G )  =  (chr
`  (Scalar `  W )
)
40 rrextchr 26369 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  (chr `  G
)  =  0 )
4129, 40syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  (chr `  G )  =  0 )
4239, 41syl5eqr 2487 . . . . . . . . . . 11  |-  ( ph  ->  (chr `  (Scalar `  W
) )  =  0 )
43 rrextcusp 26370 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  G  e. CUnifSp )
4429, 43syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  G  e. CUnifSp )
4518, 44syl5eqelr 2526 . . . . . . . . . . 11  |-  ( ph  ->  (Scalar `  W )  e. CUnifSp )
4618fveq2i 5691 . . . . . . . . . . . 12  |-  (UnifSt `  G )  =  (UnifSt `  (Scalar `  W )
)
47 eqid 2441 . . . . . . . . . . . . . 14  |-  ( Base `  G )  =  (
Base `  G )
4847, 17rrextust 26373 . . . . . . . . . . . . 13  |-  ( G  e. ℝExt  ->  (UnifSt `  G )  =  (metUnif `  D )
)
4929, 48syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  (UnifSt `  G )  =  (metUnif `  D )
)
5046, 49syl5eqr 2487 . . . . . . . . . . 11  |-  ( ph  ->  (UnifSt `  (Scalar `  W
) )  =  (metUnif `  D ) )
5123, 24, 25, 26, 27, 32, 35, 38, 42, 45, 50rrhf 26363 . . . . . . . . . 10  |-  ( ph  ->  (RRHom `  (Scalar `  W
) ) : RR --> ( Base `  (Scalar `  W
) ) )
526feq1i 5548 . . . . . . . . . 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 5558 . . . . . . . . 9  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  Fun  H )
5553, 54syl 16 . . . . . . . 8  |-  ( ph  ->  Fun  H )
56 0re 9382 . . . . . . . . . . 11  |-  0  e.  RR
57 pnfxr 11088 . . . . . . . . . . 11  |- +oo  e.  RR*
58 icossre 11372 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
5956, 57, 58mp2an 667 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
6059a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( 0 [,) +oo )  C_  RR )
61 fdm 5560 . . . . . . . . . 10  |-  ( H : RR --> ( Base `  (Scalar `  W )
)  ->  dom  H  =  RR )
6253, 61syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  H  =  RR )
6360, 62sseqtr4d 3390 . . . . . . . 8  |-  ( ph  ->  ( 0 [,) +oo )  C_  dom  H )
64 funfvima2 5950 . . . . . . . 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 656 . . . . . . 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 656 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( H `  ( M `  ( `' F " { x }
) ) )  e.  ( H " (
0 [,) +oo )
) )
68 isrnmeas 26550 . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 5698 . . . . . . . . . . . . . . 15  |-  ( TopOpen `  W )  e.  _V
722, 71eqeltri 2511 . . . . . . . . . . . . . 14  |-  J  e. 
_V
7372a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  _V )
7473sgsiga 26521 . . . . . . . . . . . 12  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
753, 74syl5eqel 2525 . . . . . . . . . . 11  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
761, 2, 3, 4, 5, 6, 7, 8, 9sibfmbl 26651 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( dom 
MMblFnM S ) )
7770, 75, 76mbfmf 26606 . . . . . . . . . 10  |-  ( ph  ->  F : U. dom  M --> U. S )
78 frn 5562 . . . . . . . . . 10  |-  ( F : U. dom  M --> U. S  ->  ran  F  C_ 
U. S )
7977, 78syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  U. S
)
803unieqi 4097 . . . . . . . . . . 11  |-  U. S  =  U. (sigaGen `  J
)
81 unisg 26522 . . . . . . . . . . . 12  |-  ( J  e.  _V  ->  U. (sigaGen `  J )  =  U. J )
8272, 81mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->  U. (sigaGen `  J
)  =  U. J
)
8380, 82syl5eq 2485 . . . . . . . . . 10  |-  ( ph  ->  U. S  =  U. J )
84 sitgclg.1 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  TopSp )
851, 2tpsuni 18502 . . . . . . . . . . 11  |-  ( W  e.  TopSp  ->  B  =  U. J )
8684, 85syl 16 . . . . . . . . . 10  |-  ( ph  ->  B  =  U. J
)
8783, 86eqtr4d 2476 . . . . . . . . 9  |-  ( ph  ->  U. S  =  B )
8879, 87sseqtrd 3389 . . . . . . . 8  |-  ( ph  ->  ran  F  C_  B
)
8988ssdifd 3489 . . . . . . 7  |-  ( ph  ->  ( ran  F  \  {  .0.  } )  C_  ( B  \  {  .0.  } ) )
9089sselda 3353 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  ( B  \  {  .0.  } ) )
9190eldifad 3337 . . . . 5  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  ->  x  e.  B )
92 simp2 984 . . . . . 6  |-  ( (
ph  /\  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) )  /\  x  e.  B )  ->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H " ( 0 [,) +oo ) ) )
93 eleq1 2501 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  e.  ( H " (
0 [,) +oo )
)  <->  ( H `  ( M `  ( `' F " { x } ) ) )  e.  ( H "
( 0 [,) +oo ) ) ) )
94933anbi2d 1289 . . . . . . . 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 6097 . . . . . . . . 9  |-  ( m  =  ( H `  ( M `  ( `' F " { x } ) ) )  ->  ( m  .x.  x )  =  ( ( H `  ( M `  ( `' F " { x }
) ) )  .x.  x ) )
9695eleq1d 2507 . . . . . . . 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 3027 . . . . . 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 1213 . . . 4  |-  ( (
ph  /\  x  e.  ( ran  F  \  {  .0.  } ) )  -> 
( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x )  e.  B )
102 eqid 2441 . . . 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 5864 . . 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 26653 . . . 4  |-  ( ph  ->  ran  F  e.  Fin )
105 cnvimass 5186 . . . . . . 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 5331 . . . . . . 7  |-  dom  (
x  e.  ( ran 
F  \  {  .0.  } )  |->  ( ( H `
 ( M `  ( `' F " { x } ) ) ) 
.x.  x ) ) 
C_  ( ran  F  \  {  .0.  } )
107105, 106sstri 3362 . . . . . 6  |-  ( `' ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) " ( _V  \  {  .0.  }
) )  C_  ( ran  F  \  {  .0.  } )
108 difss 3480 . . . . . 6  |-  ( ran 
F  \  {  .0.  } )  C_  ran  F
109107, 108sstri 3362 . . . . 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 7529 . . . 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 656 . . 3  |-  ( ph  ->  ( `' ( x  e.  ( ran  F  \  {  .0.  } ) 
|->  ( ( H `  ( M `  ( `' F " { x } ) ) ) 
.x.  x ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
1131, 4, 11, 14, 103, 112gsumclOLD 16393 . 2  |-  ( ph  ->  ( W  gsumg  ( x  e.  ( ran  F  \  {  .0.  } )  |->  ( ( H `  ( M `
 ( `' F " { x } ) ) )  .x.  x
) ) )  e.  B )
11410, 113eqeltrd 2515 1  |-  ( ph  ->  ( ( Wsitg M
) `  F )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    \ cdif 3322    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   {csn 3874   U.cuni 4088  Disj wdisj 4259   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   `'ccnv 4835   dom cdm 4836   ran crn 4837    |` cres 4838   "cima 4839   Fun wfun 5409   -->wf 5411   ` cfv 5415  (class class class)co 6090   omcom 6475    ~<_ cdom 7304   Fincfn 7306   RRcr 9277   0cc0 9278   +oocpnf 9411   RR*cxr 9413   (,)cioo 11296   [,)cico 11298   [,]cicc 11299   Basecbs 14170  Scalarcsca 14237   .scvsca 14238   distcds 14243   TopOpenctopn 14356   topGenctg 14372   0gc0g 14374    gsumg cgsu 14375  CMndccmn 16270   DivRingcdr 16812  metUnifcmetu 17767   ZModczlm 17891  chrcchr 17892   TopSpctps 18460  UnifStcuss 19787  CUnifSpccusp 19831  NrmRingcnrg 20131  NrmModcnlm 20132  RRHomcrrh 26358   ℝExt crrext 26359  Σ*cesum 26419  sigAlgebracsiga 26486  sigaGencsigagen 26517  measurescmeas 26545  sitgcsitg 26645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-numer 13809  df-denom 13810  df-gz 13987  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-ghm 15738  df-cntz 15828  df-od 16025  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-rnghom 16796  df-drng 16814  df-subrg 16843  df-abv 16882  df-lmod 16930  df-nzr 17318  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-metu 17776  df-cnfld 17778  df-zring 17843  df-zrh 17894  df-zlm 17895  df-chr 17896  df-refld 17994  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-cn 18790  df-cnp 18791  df-haus 18878  df-reg 18879  df-cmp 18949  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-fcls 19473  df-cnext 19591  df-ust 19734  df-utop 19765  df-uss 19790  df-usp 19791  df-ucn 19810  df-cfilu 19821  df-cusp 19832  df-xms 19854  df-ms 19855  df-tms 19856  df-nm 20134  df-ngp 20135  df-nrg 20137  df-nlm 20138  df-cncf 20413  df-cfil 20725  df-cmet 20727  df-cms 20805  df-qqh 26338  df-rrh 26360  df-rrext 26364  df-esum 26420  df-siga 26487  df-sigagen 26518  df-meas 26546  df-mbfm 26602  df-sitg 26646
This theorem is referenced by:  sitgclbn  26659
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