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Theorem imasbas 15369
Description: The base set of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
imasbas.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasbas.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasbas.f  |-  ( ph  ->  F : V -onto-> B
)
imasbas.r  |-  ( ph  ->  R  e.  Z )
Assertion
Ref Expression
imasbas  |-  ( ph  ->  B  =  ( Base `  U ) )

Proof of Theorem imasbas
Dummy variables  g  h  i  n  p  q  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . . 4  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasbas.v . . . 4  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2429 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2429 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2429 . . . 4  |-  (Scalar `  R )  =  (Scalar `  R )
6 eqid 2429 . . . 4  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 eqid 2429 . . . 4  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2429 . . . 4  |-  ( .i
`  R )  =  ( .i `  R
)
9 eqid 2429 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
10 eqid 2429 . . . 4  |-  ( dist `  R )  =  (
dist `  R )
11 eqid 2429 . . . 4  |-  ( le
`  R )  =  ( le `  R
)
12 eqidd 2430 . . . 4  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( +g  `  R ) q ) ) >. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } )
13 eqidd 2430 . . . 4  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } )
14 eqidd 2430 . . . 4  |-  ( ph  ->  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) )  =  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R ) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s
`  R ) q ) ) ) )
15 eqidd 2430 . . . 4  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p ( .i `  R ) q )
>. } )
16 eqidd 2430 . . . 4  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
17 eqidd 2430 . . . 4  |-  ( ph  ->  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( ( dist `  R
)  o.  g ) ) ) ,  RR* ,  `'  <  ) )  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( ( dist `  R
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) )
18 eqidd 2430 . . . 4  |-  ( ph  ->  ( ( F  o.  ( le `  R ) )  o.  `' F
)  =  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )
19 imasbas.f . . . 4  |-  ( ph  ->  F : V -onto-> B
)
20 imasbas.r . . . 4  |-  ( ph  ->  R  e.  Z )
211, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20imasval 15368 . . 3  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( ( dist `  R
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } ) )
22 eqid 2429 . . . 4  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( ( dist `  R
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } )  =  ( ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( +g  `  R ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( ( dist `  R
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } )
2322imasvalstr 15309 . . 3  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( ( dist `  R
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } ) Struct  <. 1 , ; 1
2 >.
24 baseid 15132 . . 3  |-  Base  = Slot  ( Base `  ndx )
25 snsstp1 4154 . . . . 5  |-  { <. (
Base `  ndx ) ,  B >. }  C_  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( +g  `  R ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }
26 ssun1 3635 . . . . 5  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( +g  `  R ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. } 
C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( +g  `  R ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )
2725, 26sstri 3479 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. }  C_  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )
28 ssun1 3635 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p ( +g  `  R
) q ) )
>. } >. ,  <. ( .r `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )  C_  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( +g  `  R ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( ( dist `  R
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } )
2927, 28sstri 3479 . . 3  |-  { <. (
Base `  ndx ) ,  B >. }  C_  (
( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( +g  `  R ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p ( .r
`  R ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  R ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  R
) ) ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p ( .s `  R
) q ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p
( .i `  R
) q ) >. } >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  R ) qTop  F
) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR*s  gsumg  ( ( dist `  R
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } )
30 fvex 5891 . . . . 5  |-  ( Base `  R )  e.  _V
312, 30syl6eqel 2525 . . . 4  |-  ( ph  ->  V  e.  _V )
32 fornex 6776 . . . 4  |-  ( V  e.  _V  ->  ( F : V -onto-> B  ->  B  e.  _V )
)
3331, 19, 32sylc 62 . . 3  |-  ( ph  ->  B  e.  _V )
34 eqid 2429 . . 3  |-  ( Base `  U )  =  (
Base `  U )
3521, 23, 24, 29, 33, 34strfv3 15121 . 2  |-  ( ph  ->  ( Base `  U
)  =  B )
3635eqcomd 2437 1  |-  ( ph  ->  B  =  ( Base `  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   _Vcvv 3087    u. cun 3440   {csn 4002   {ctp 4006   <.cop 4008   U_ciun 4302    |-> cmpt 4484    X. cxp 4852   `'ccnv 4853   ran crn 4855    o. ccom 4858   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806    ^m cmap 7480   supcsup 7960   1c1 9539    + caddc 9541   RR*cxr 9673    < clt 9674    - cmin 9859   NNcn 10609   2c2 10659  ;cdc 11051   ...cfz 11782   ndxcnx 15081   Basecbs 15084   +g cplusg 15152   .rcmulr 15153  Scalarcsca 15155   .scvsca 15156   .icip 15157  TopSetcts 15158   lecple 15159   distcds 15161   TopOpenctopn 15279    gsumg cgsu 15298   RR*scxrs 15357   qTop cqtop 15360    "s cimas 15361
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-imas 15365
This theorem is referenced by:  qusbas  15402  xpsbas  15431  imasmnd2  16524  imasgrp2  16752  imasring  17782  imastopn  20666  imastps  20667  imasf1oxms  21435  imasf1oms  21436  circtopn  28503  imasgim  35664  isnumbasgrplem1  35666
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