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Theorem sitmcl 24616
Description: Closure of the integral distance between two simple functions, for an extended metric space. (Contributed by Thierry Arnoux, 13-Feb-2018.)
Hypotheses
Ref Expression
sitmcl.0  |-  ( ph  ->  W  e.  Mnd )
sitmcl.1  |-  ( ph  ->  W  e.  * MetSp )
sitmcl.2  |-  ( ph  ->  M  e.  U. ran measures )
sitmcl.3  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
sitmcl.4  |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )
Assertion
Ref Expression
sitmcl  |-  ( ph  ->  ( F ( Wsitm M ) G )  e.  ( 0 [,] 
+oo ) )

Proof of Theorem sitmcl
Dummy variables  x  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( dist `  W )  =  (
dist `  W )
2 sitmcl.1 . . 3  |-  ( ph  ->  W  e.  * MetSp )
3 sitmcl.2 . . 3  |-  ( ph  ->  M  e.  U. ran measures )
4 sitmcl.3 . . 3  |-  ( ph  ->  F  e.  dom  ( Wsitg M ) )
5 sitmcl.4 . . 3  |-  ( ph  ->  G  e.  dom  ( Wsitg M ) )
61, 2, 3, 4, 5sitmfval 24615 . 2  |-  ( ph  ->  ( F ( Wsitm M ) G )  =  ( ( (
RR* ss  ( 0 [,] 
+oo ) )sitg M
) `  ( F  o F ( dist `  W
) G ) ) )
7 xrge0base 24160 . . 3  |-  ( 0 [,]  +oo )  =  (
Base `  ( RR* ss  ( 0 [,]  +oo ) ) )
8 xrge0topn 24282 . . . 4  |-  ( TopOpen `  ( RR* ss  ( 0 [,] 
+oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )
98eqcomi 2408 . . 3  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  =  ( TopOpen `  ( RR* ss  ( 0 [,]  +oo ) ) )
10 eqid 2404 . . 3  |-  (sigaGen `  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) ) )  =  (sigaGen `  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) ) )
11 xrge00 24161 . . 3  |-  0  =  ( 0g `  ( RR* ss  ( 0 [,] 
+oo ) ) )
12 ovex 6065 . . . 4  |-  ( 0 [,]  +oo )  e.  _V
13 eqid 2404 . . . . 5  |-  ( RR* ss  ( 0 [,]  +oo ) )  =  (
RR* ss  ( 0 [,] 
+oo ) )
14 ax-xrsvsca 24149 . . . . 5  |-  x e  =  ( .s `  RR* s )
1513, 14ressvsca 13560 . . . 4  |-  ( ( 0 [,]  +oo )  e.  _V  ->  x e  =  ( .s `  ( RR* ss  ( 0 [,] 
+oo ) ) ) )
1612, 15ax-mp 8 . . 3  |-  x e  =  ( .s `  ( RR* ss  ( 0 [,] 
+oo ) ) )
17 ax-xrssca 24148 . . . . . 6  |-  (flds  RR )  =  (Scalar `  RR* s )
1813, 17resssca 13559 . . . . 5  |-  ( ( 0 [,]  +oo )  e.  _V  ->  (flds  RR )  =  (Scalar `  ( RR* ss  ( 0 [,]  +oo ) ) ) )
1912, 18ax-mp 8 . . . 4  |-  (flds  RR )  =  (Scalar `  ( RR* ss  ( 0 [,]  +oo ) ) )
2019fveq2i 5690 . . 3  |-  (RRHom `  (flds  RR ) )  =  (RRHom `  (Scalar `  ( RR* ss  ( 0 [,]  +oo ) ) ) )
21 ovex 6065 . . . 4  |-  ( RR* ss  ( 0 [,]  +oo ) )  e.  _V
2221a1i 11 . . 3  |-  ( ph  ->  ( RR* ss  ( 0 [,]  +oo ) )  e. 
_V )
23 eqid 2404 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
24 eqid 2404 . . . . . . 7  |-  ( TopOpen `  W )  =  (
TopOpen `  W )
25 eqid 2404 . . . . . . 7  |-  (sigaGen `  ( TopOpen
`  W ) )  =  (sigaGen `  ( TopOpen
`  W ) )
26 eqid 2404 . . . . . . 7  |-  ( 0g
`  W )  =  ( 0g `  W
)
27 eqid 2404 . . . . . . 7  |-  ( .s
`  W )  =  ( .s `  W
)
28 eqid 2404 . . . . . . 7  |-  (RRHom `  (Scalar `  W ) )  =  (RRHom `  (Scalar `  W ) )
2923, 24, 25, 26, 27, 28, 2, 3, 4sibff 24604 . . . . . 6  |-  ( ph  ->  F : U. dom  M --> U. ( TopOpen `  W
) )
30 xmstps 18436 . . . . . . . 8  |-  ( W  e.  * MetSp  ->  W  e.  TopSp )
3123, 24tpsuni 16958 . . . . . . . 8  |-  ( W  e.  TopSp  ->  ( Base `  W )  =  U. ( TopOpen `  W )
)
322, 30, 313syl 19 . . . . . . 7  |-  ( ph  ->  ( Base `  W
)  =  U. ( TopOpen
`  W ) )
33 feq3 5537 . . . . . . 7  |-  ( (
Base `  W )  =  U. ( TopOpen `  W
)  ->  ( F : U. dom  M --> ( Base `  W )  <->  F : U. dom  M --> U. ( TopOpen
`  W ) ) )
3432, 33syl 16 . . . . . 6  |-  ( ph  ->  ( F : U. dom  M --> ( Base `  W
)  <->  F : U. dom  M --> U. ( TopOpen `  W
) ) )
3529, 34mpbird 224 . . . . 5  |-  ( ph  ->  F : U. dom  M --> ( Base `  W
) )
3623, 24, 25, 26, 27, 28, 2, 3, 5sibff 24604 . . . . . 6  |-  ( ph  ->  G : U. dom  M --> U. ( TopOpen `  W
) )
37 feq3 5537 . . . . . . 7  |-  ( (
Base `  W )  =  U. ( TopOpen `  W
)  ->  ( G : U. dom  M --> ( Base `  W )  <->  G : U. dom  M --> U. ( TopOpen
`  W ) ) )
3832, 37syl 16 . . . . . 6  |-  ( ph  ->  ( G : U. dom  M --> ( Base `  W
)  <->  G : U. dom  M --> U. ( TopOpen `  W
) ) )
3936, 38mpbird 224 . . . . 5  |-  ( ph  ->  G : U. dom  M --> ( Base `  W
) )
40 dmexg 5089 . . . . . 6  |-  ( M  e.  U. ran measures  ->  dom  M  e.  _V )
41 uniexg 4665 . . . . . 6  |-  ( dom 
M  e.  _V  ->  U.
dom  M  e.  _V )
423, 40, 413syl 19 . . . . 5  |-  ( ph  ->  U. dom  M  e. 
_V )
4335, 39, 42ofresid 24008 . . . 4  |-  ( ph  ->  ( F  o F ( dist `  W
) G )  =  ( F  o F ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) ) G ) )
442, 30syl 16 . . . . 5  |-  ( ph  ->  W  e.  TopSp )
45 eqid 2404 . . . . . . . 8  |-  ( (
dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) )  =  ( ( dist `  W )  |`  (
( Base `  W )  X.  ( Base `  W
) ) )
4623, 45xmsxmet 18439 . . . . . . 7  |-  ( W  e.  * MetSp  ->  (
( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) )  e.  ( * Met `  ( Base `  W
) ) )
47 xmetpsmet 18331 . . . . . . 7  |-  ( ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) )  e.  ( * Met `  ( Base `  W ) )  -> 
( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) )  e.  (PsMet `  ( Base `  W )
) )
482, 46, 473syl 19 . . . . . 6  |-  ( ph  ->  ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) )  e.  (PsMet `  ( Base `  W )
) )
49 psmetxrge0 18297 . . . . . 6  |-  ( ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) )  e.  (PsMet `  ( Base `  W )
)  ->  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) ) : ( ( Base `  W )  X.  ( Base `  W ) ) --> ( 0 [,]  +oo ) )
5048, 49syl 16 . . . . 5  |-  ( ph  ->  ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) ) : ( (
Base `  W )  X.  ( Base `  W
) ) --> ( 0 [,]  +oo ) )
51 xrge0tps 24281 . . . . . 6  |-  ( RR* ss  ( 0 [,]  +oo ) )  e.  TopSp
5251a1i 11 . . . . 5  |-  ( ph  ->  ( RR* ss  ( 0 [,]  +oo ) )  e. 
TopSp )
5324, 23, 45xmstopn 18434 . . . . . . . 8  |-  ( W  e.  * MetSp  ->  ( TopOpen
`  W )  =  ( MetOpen `  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) ) ) )
542, 53syl 16 . . . . . . 7  |-  ( ph  ->  ( TopOpen `  W )  =  ( MetOpen `  (
( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) ) ) )
55 eqid 2404 . . . . . . . . 9  |-  ( MetOpen `  ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) ) )  =  (
MetOpen `  ( ( dist `  W )  |`  (
( Base `  W )  X.  ( Base `  W
) ) ) )
5655methaus 18503 . . . . . . . 8  |-  ( ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) )  e.  ( * Met `  ( Base `  W ) )  -> 
( MetOpen `  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) ) )  e.  Haus )
572, 46, 563syl 19 . . . . . . 7  |-  ( ph  ->  ( MetOpen `  ( ( dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) ) )  e.  Haus )
5854, 57eqeltrd 2478 . . . . . 6  |-  ( ph  ->  ( TopOpen `  W )  e.  Haus )
59 haust1 17370 . . . . . 6  |-  ( (
TopOpen `  W )  e. 
Haus  ->  ( TopOpen `  W
)  e.  Fre )
6058, 59syl 16 . . . . 5  |-  ( ph  ->  ( TopOpen `  W )  e.  Fre )
612, 46syl 16 . . . . . . 7  |-  ( ph  ->  ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) )  e.  ( * Met `  ( Base `  W ) ) )
62 sitmcl.0 . . . . . . . 8  |-  ( ph  ->  W  e.  Mnd )
6323, 26mndidcl 14669 . . . . . . . 8  |-  ( W  e.  Mnd  ->  ( 0g `  W )  e.  ( Base `  W
) )
6462, 63syl 16 . . . . . . 7  |-  ( ph  ->  ( 0g `  W
)  e.  ( Base `  W ) )
65 xmet0 18325 . . . . . . 7  |-  ( ( ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) )  e.  ( * Met `  ( Base `  W ) )  /\  ( 0g `  W )  e.  ( Base `  W
) )  ->  (
( 0g `  W
) ( ( dist `  W )  |`  (
( Base `  W )  X.  ( Base `  W
) ) ) ( 0g `  W ) )  =  0 )
6661, 64, 65syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( 0g `  W ) ( (
dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) ) ( 0g `  W
) )  =  0 )
6766, 11syl6eq 2452 . . . . 5  |-  ( ph  ->  ( ( 0g `  W ) ( (
dist `  W )  |`  ( ( Base `  W
)  X.  ( Base `  W ) ) ) ( 0g `  W
) )  =  ( 0g `  ( RR* ss  ( 0 [,]  +oo ) ) ) )
6823, 24, 25, 26, 27, 28, 2, 3, 4, 7, 44, 50, 5, 52, 60, 67sibfof 24607 . . . 4  |-  ( ph  ->  ( F  o F ( ( dist `  W
)  |`  ( ( Base `  W )  X.  ( Base `  W ) ) ) G )  e. 
dom  ( ( RR* ss  ( 0 [,]  +oo ) )sitg M ) )
6943, 68eqeltrd 2478 . . 3  |-  ( ph  ->  ( F  o F ( dist `  W
) G )  e. 
dom  ( ( RR* ss  ( 0 [,]  +oo ) )sitg M ) )
70 eqid 2404 . . . . . 6  |-  (flds  RR )  =  (flds  RR )
7170rebase 24222 . . . . 5  |-  RR  =  ( Base `  (flds  RR ) )
7271, 71xpeq12i 4859 . . . 4  |-  ( RR 
X.  RR )  =  ( ( Base `  (flds  RR )
)  X.  ( Base `  (flds  RR ) ) )
7372reseq2i 5102 . . 3  |-  ( (
dist `  (flds  RR ) )  |`  ( RR  X.  RR ) )  =  ( ( dist `  (flds  RR )
)  |`  ( ( Base `  (flds  RR ) )  X.  ( Base `  (flds  RR ) ) ) )
74 xrge0cmn 16695 . . . 4  |-  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
7574a1i 11 . . 3  |-  ( ph  ->  ( RR* ss  ( 0 [,]  +oo ) )  e. CMnd
)
7670refld 24232 . . . . . 6  |-  (flds  RR )  e. Field
77 isfld 15799 . . . . . 6  |-  ( (flds  RR )  e. Field 
<->  ( (flds  RR )  e.  DivRing  /\  (flds  RR )  e.  CRing ) )
7876, 77mpbi 200 . . . . 5  |-  ( (flds  RR )  e.  DivRing  /\  (flds  RR )  e.  CRing )
7978simpli 445 . . . 4  |-  (flds  RR )  e.  DivRing
8079a1i 11 . . 3  |-  ( ph  ->  (flds  RR )  e.  DivRing )
81 cnnrg 18768 . . . . 5  |-fld  e. NrmRing
82 resubdrg 16705 . . . . . 6  |-  ( RR  e.  (SubRing ` fld )  /\  (flds  RR )  e.  DivRing )
8382simpli 445 . . . . 5  |-  RR  e.  (SubRing ` fld )
8470subrgnrg 18662 . . . . 5  |-  ( (fld  e. NrmRing  /\  RR  e.  (SubRing ` fld ) )  ->  (flds  RR )  e. NrmRing )
8581, 83, 84mp2an 654 . . . 4  |-  (flds  RR )  e. NrmRing
8685a1i 11 . . 3  |-  ( ph  ->  (flds  RR )  e. NrmRing )
8770rezh 24308 . . . 4  |-  ( ZMod
`  (flds  RR ) )  e. NrmMod
8887a1i 11 . . 3  |-  ( ph  ->  ( ZMod `  (flds  RR )
)  e. NrmMod )
8970reofld 24233 . . . . 5  |-  (flds  RR )  e. oField
90 ofldchr 24197 . . . . 5  |-  ( (flds  RR )  e. oField  ->  (chr `  (flds  RR )
)  =  0 )
9189, 90ax-mp 8 . . . 4  |-  (chr `  (flds  RR ) )  =  0
9291a1i 11 . . 3  |-  ( ph  ->  (chr `  (flds  RR ) )  =  0 )
93 cnfldtps 18765 . . . . 5  |-fld  e.  TopSp
94 reex 9037 . . . . 5  |-  RR  e.  _V
95 resstps 17205 . . . . 5  |-  ( (fld  e. 
TopSp  /\  RR  e.  _V )  ->  (flds  RR )  e.  TopSp )
9693, 94, 95mp2an 654 . . . 4  |-  (flds  RR )  e.  TopSp
9796a1i 11 . . 3  |-  ( ph  ->  (flds  RR )  e.  TopSp )
9870recusp 24298 . . . 4  |-  (flds  RR )  e. CUnifSp
9998a1i 11 . . 3  |-  ( ph  ->  (flds  RR )  e. CUnifSp )
100 eqid 2404 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
101100cnfldtopn 18769 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
102101eqcomi 2408 . . . . . 6  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( TopOpen ` fld )
10370, 102resstopn 17204 . . . . 5  |-  ( (
MetOpen `  ( abs  o.  -  ) )t  RR )  =  ( TopOpen `  (flds  RR )
)
104102cnfldhaus 18772 . . . . . 6  |-  ( MetOpen `  ( abs  o.  -  )
)  e.  Haus
105 resthaus 17386 . . . . . 6  |-  ( ( ( MetOpen `  ( abs  o. 
-  ) )  e. 
Haus  /\  RR  e.  _V )  ->  ( ( MetOpen `  ( abs  o.  -  )
)t 
RR )  e.  Haus )
106104, 94, 105mp2an 654 . . . . 5  |-  ( (
MetOpen `  ( abs  o.  -  ) )t  RR )  e.  Haus
107103, 106eqeltrri 2475 . . . 4  |-  ( TopOpen `  (flds  RR ) )  e.  Haus
108107a1i 11 . . 3  |-  ( ph  ->  ( TopOpen `  (flds  RR ) )  e. 
Haus )
10970reust 24297 . . . 4  |-  (UnifSt `  (flds  RR ) )  =  (metUnif `  ( ( dist `  (flds  RR )
)  |`  ( RR  X.  RR ) ) )
110109a1i 11 . . 3  |-  ( ph  ->  (UnifSt `  (flds  RR ) )  =  (metUnif `  ( ( dist `  (flds  RR ) )  |`  ( RR  X.  RR ) ) ) )
111 rrhre 24340 . . . . . . . . 9  |-  (RRHom `  (flds  RR ) )  =  (  _I  |`  RR )
112111imaeq1i 5159 . . . . . . . 8  |-  ( (RRHom `  (flds  RR ) ) " (
0 [,)  +oo ) )  =  ( (  _I  |`  RR ) " (
0 [,)  +oo ) )
113 0re 9047 . . . . . . . . . 10  |-  0  e.  RR
114 pnfxr 10669 . . . . . . . . . 10  |-  +oo  e.  RR*
115 icossre 10947 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
116113, 114, 115mp2an 654 . . . . . . . . 9  |-  ( 0 [,)  +oo )  C_  RR
117 resiima 5179 . . . . . . . . 9  |-  ( ( 0 [,)  +oo )  C_  RR  ->  ( (  _I  |`  RR ) "
( 0 [,)  +oo ) )  =  ( 0 [,)  +oo )
)
118116, 117ax-mp 8 . . . . . . . 8  |-  ( (  _I  |`  RR ) " ( 0 [,) 
+oo ) )  =  ( 0 [,)  +oo )
119112, 118eqtri 2424 . . . . . . 7  |-  ( (RRHom `  (flds  RR ) ) " (
0 [,)  +oo ) )  =  ( 0 [,) 
+oo )
120 icossicc 24082 . . . . . . 7  |-  ( 0 [,)  +oo )  C_  (
0 [,]  +oo )
121119, 120eqsstri 3338 . . . . . 6  |-  ( (RRHom `  (flds  RR ) ) " (
0 [,)  +oo ) ) 
C_  ( 0 [,] 
+oo )
122121sseli 3304 . . . . 5  |-  ( m  e.  ( (RRHom `  (flds  RR ) ) " (
0 [,)  +oo ) )  ->  m  e.  ( 0 [,]  +oo )
)
1231223ad2ant2 979 . . . 4  |-  ( (
ph  /\  m  e.  ( (RRHom `  (flds  RR ) ) "
( 0 [,)  +oo ) )  /\  x  e.  ( 0 [,]  +oo ) )  ->  m  e.  ( 0 [,]  +oo ) )
124 simp3 959 . . . 4  |-  ( (
ph  /\  m  e.  ( (RRHom `  (flds  RR ) ) "
( 0 [,)  +oo ) )  /\  x  e.  ( 0 [,]  +oo ) )  ->  x  e.  ( 0 [,]  +oo ) )
125 ge0xmulcl 10968 . . . 4  |-  ( ( m  e.  ( 0 [,]  +oo )  /\  x  e.  ( 0 [,]  +oo ) )  ->  (
m x e x )  e.  ( 0 [,]  +oo ) )
126123, 124, 125syl2anc 643 . . 3  |-  ( (
ph  /\  m  e.  ( (RRHom `  (flds  RR ) ) "
( 0 [,)  +oo ) )  /\  x  e.  ( 0 [,]  +oo ) )  ->  (
m x e x )  e.  ( 0 [,]  +oo ) )
1277, 9, 10, 11, 16, 20, 22, 3, 69, 19, 73, 52, 75, 80, 86, 88, 92, 97, 99, 108, 110, 126sitgclg 24609 . 2  |-  ( ph  ->  ( ( ( RR* ss  ( 0 [,]  +oo ) )sitg M ) `  ( F  o F
( dist `  W ) G ) )  e.  ( 0 [,]  +oo ) )
1286, 127eqeltrd 2478 1  |-  ( ph  ->  ( F ( Wsitm M ) G )  e.  ( 0 [,] 
+oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   U.cuni 3975    _I cid 4453    X. cxp 4835   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   RRcr 8945   0cc0 8946    +oocpnf 9073   RR*cxr 9075    <_ cle 9077    - cmin 9247   x ecxmu 10665   [,)cico 10874   [,]cicc 10875   abscabs 11994   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487   .scvsca 13488   distcds 13493   ↾t crest 13603   TopOpenctopn 13604  ordTopcordt 13676   RR* scxrs 13677   0gc0g 13678   Mndcmnd 14639  CMndccmn 15367   CRingccrg 15616   DivRingcdr 15790  Fieldcfield 15791  SubRingcsubrg 15819  PsMetcpsmet 16640   * Metcxmt 16641   MetOpencmopn 16646  metUnifcmetu 16648  ℂfldccnfld 16658   ZModczlm 16734  chrcchr 16735   TopSpctps 16916   Frect1 17325   Hauscha 17326  UnifStcuss 18236  CUnifSpccusp 18280   *
MetSpcxme 18300  NrmRingcnrg 18580  NrmModcnlm 18581  oFieldcofld 24186  RRHomcrrh 24330  sigaGencsigagen 24474  measurescmeas 24502  sitmcsitm 24596  sitgcsitg 24597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-ac2 8299  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-xrssca 24148  ax-xrsvsca 24149
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-ac 7953  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-ordt 13680  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-poset 14358  df-plt 14370  df-toset 14418  df-ps 14584  df-tsr 14585  df-mnd 14645  df-plusf 14646  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-od 15122  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-field 15793  df-subrg 15821  df-abv 15860  df-lmod 15907  df-scaf 15908  df-sra 16199  df-rgmod 16200  df-nzr 16284  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-metu 16657  df-cnfld 16659  df-zrh 16737  df-zlm 16738  df-chr 16739  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-t1 17332  df-haus 17333  df-reg 17334  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-fcls 17926  df-cnext 18044  df-tmd 18055  df-tgp 18056  df-tsms 18109  df-trg 18142  df-ust 18183  df-utop 18214  df-uss 18239  df-usp 18240  df-ucn 18259  df-cfilu 18270  df-cusp 18281  df-xms 18303  df-ms 18304  df-tms 18305  df-nm 18583  df-ngp 18584  df-nrg 18586  df-nlm 18587  df-ii 18860  df-cncf 18861  df-cfil 19161  df-cmet 19163  df-cms 19241  df-limc 19706  df-dv 19707  df-log 20407  df-ofld 24187  df-qqh 24310  df-rrh 24331  df-esum 24378  df-siga 24444  df-sigagen 24475  df-meas 24503  df-mbfm 24554  df-sitg 24598  df-sitm 24599
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