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Theorem prdsle 14405
Description: Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
prdsbas.p  |-  P  =  ( S X_s R )
prdsbas.s  |-  ( ph  ->  S  e.  V )
prdsbas.r  |-  ( ph  ->  R  e.  W )
prdsbas.b  |-  B  =  ( Base `  P
)
prdsbas.i  |-  ( ph  ->  dom  R  =  I )
prdsle.l  |-  .<_  =  ( le `  P )
Assertion
Ref Expression
prdsle  |-  ( ph  -> 
.<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
Distinct variable groups:    f, g, x, B    ph, f, g, x    f, I, g, x    P, f, g, x    R, f, g, x    S, f, g, x
Allowed substitution hints:    .<_ ( x, f,
g)    V( x, f, g)    W( x, f, g)

Proof of Theorem prdsle
Dummy variables  a 
c  d  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . 3  |-  P  =  ( S X_s R )
2 eqid 2443 . . 3  |-  ( Base `  S )  =  (
Base `  S )
3 prdsbas.i . . 3  |-  ( ph  ->  dom  R  =  I )
4 prdsbas.s . . . 4  |-  ( ph  ->  S  e.  V )
5 prdsbas.r . . . 4  |-  ( ph  ->  R  e.  W )
6 prdsbas.b . . . 4  |-  B  =  ( Base `  P
)
71, 4, 5, 6, 3prdsbas 14400 . . 3  |-  ( ph  ->  B  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
8 eqid 2443 . . . 4  |-  ( +g  `  P )  =  ( +g  `  P )
91, 4, 5, 6, 3, 8prdsplusg 14401 . . 3  |-  ( ph  ->  ( +g  `  P
)  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x )
) ( g `  x ) ) ) ) )
10 eqid 2443 . . . 4  |-  ( .r
`  P )  =  ( .r `  P
)
111, 4, 5, 6, 3, 10prdsmulr 14402 . . 3  |-  ( ph  ->  ( .r `  P
)  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r `  ( R `  x )
) ( g `  x ) ) ) ) )
12 eqid 2443 . . . 4  |-  ( .s
`  P )  =  ( .s `  P
)
131, 4, 5, 6, 3, 2, 12prdsvsca 14403 . . 3  |-  ( ph  ->  ( .s `  P
)  =  ( f  e.  ( Base `  S
) ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `
 x ) ) ( g `  x
) ) ) ) )
14 eqidd 2444 . . 3  |-  ( ph  ->  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i `  ( R `  x )
) ( g `  x ) ) ) ) ) )
15 eqidd 2444 . . 3  |-  ( ph  ->  ( Xt_ `  ( TopOpen  o.  R ) )  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
16 eqidd 2444 . . 3  |-  ( ph  ->  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
17 eqidd 2444 . . 3  |-  ( ph  ->  ( f  e.  B ,  g  e.  B  |->  sup ( ( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )  =  ( f  e.  B ,  g  e.  B  |->  sup ( ( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) )
18 eqidd 2444 . . 3  |-  ( ph  ->  ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) )  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) )
19 eqidd 2444 . . 3  |-  ( ph  ->  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) ( 2nd `  a
) ) ,  e  e.  ( ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) )  =  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) ( 2nd `  a
) ) ,  e  e.  ( ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) )
201, 2, 3, 7, 9, 11, 13, 14, 15, 16, 17, 18, 19, 4, 5prdsval 14398 . 2  |-  ( ph  ->  P  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  P ) >. ,  <. ( .r `  ndx ) ,  ( .r `  P ) >. }  u.  {
<. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( .s `  P
) >. ,  <. ( .i `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) )
>. } )  u.  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  R )
) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  B , 
g  e.  B  |->  sup ( ( ran  (
x  e.  I  |->  ( ( f `  x
) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) ( 2nd `  a
) ) ,  e  e.  ( ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } ) ) )
21 prdsle.l . 2  |-  .<_  =  ( le `  P )
22 pleid 14338 . 2  |-  le  = Slot  ( le `  ndx )
23 fvex 5706 . . . . . 6  |-  ( Base `  P )  e.  _V
246, 23eqeltri 2513 . . . . 5  |-  B  e. 
_V
2524, 24xpex 6513 . . . 4  |-  ( B  X.  B )  e. 
_V
26 vex 2980 . . . . . . . 8  |-  f  e. 
_V
27 vex 2980 . . . . . . . 8  |-  g  e. 
_V
2826, 27prss 4032 . . . . . . 7  |-  ( ( f  e.  B  /\  g  e.  B )  <->  { f ,  g } 
C_  B )
2928anbi1i 695 . . . . . 6  |-  ( ( ( f  e.  B  /\  g  e.  B
)  /\  A. x  e.  I  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) )  <->  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) )
3029opabbii 4361 . . . . 5  |-  { <. f ,  g >.  |  ( ( f  e.  B  /\  g  e.  B
)  /\  A. x  e.  I  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) }  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) }
31 opabssxp 4916 . . . . 5  |-  { <. f ,  g >.  |  ( ( f  e.  B  /\  g  e.  B
)  /\  A. x  e.  I  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) }  C_  ( B  X.  B )
3230, 31eqsstr3i 3392 . . . 4  |-  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } 
C_  ( B  X.  B )
3325, 32ssexi 4442 . . 3  |-  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) }  e.  _V
3433a1i 11 . 2  |-  ( ph  ->  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }  e.  _V )
35 snsstp2 4030 . . . 4  |-  { <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. }  C_  { <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( TopOpen  o.  R
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  B , 
g  e.  B  |->  sup ( ( ran  (
x  e.  I  |->  ( ( f `  x
) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }
36 ssun1 3524 . . . 4  |-  { <. (TopSet `  ndx ) ,  (
Xt_ `  ( TopOpen  o.  R
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  B , 
g  e.  B  |->  sup ( ( ran  (
x  e.  I  |->  ( ( f `  x
) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  C_  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  R )
) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  B , 
g  e.  B  |->  sup ( ( ran  (
x  e.  I  |->  ( ( f `  x
) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) ( 2nd `  a
) ) ,  e  e.  ( ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } )
3735, 36sstri 3370 . . 3  |-  { <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. }  C_  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  R ) )
>. ,  <. ( le
`  ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) } >. ,  <. (
dist `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  sup ( ( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) ( 2nd `  a
) ) ,  e  e.  ( ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } )
38 ssun2 3525 . . 3  |-  ( {
<. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  R ) )
>. ,  <. ( le
`  ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) } >. ,  <. (
dist `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  sup ( ( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) ( 2nd `  a
) ) ,  e  e.  ( ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } ) 
C_  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  P ) >. ,  <. ( .r `  ndx ) ,  ( .r `  P ) >. }  u.  {
<. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( .s `  P
) >. ,  <. ( .i `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) )
>. } )  u.  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  R )
) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  B , 
g  e.  B  |->  sup ( ( ran  (
x  e.  I  |->  ( ( f `  x
) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) ( 2nd `  a
) ) ,  e  e.  ( ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } ) )
3937, 38sstri 3370 . 2  |-  { <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. }  C_  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  P ) >. ,  <. ( .r `  ndx ) ,  ( .r `  P ) >. }  u.  {
<. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  ( .s `  P
) >. ,  <. ( .i `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) )
>. } )  u.  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  R )
) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  B , 
g  e.  B  |->  sup ( ( ran  (
x  e.  I  |->  ( ( f `  x
) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. ( Hom  `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c ( f  e.  B ,  g  e.  B  |-> 
X_ x  e.  I 
( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) ) ( 2nd `  a
) ) ,  e  e.  ( ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
( Hom  `  ( R `
 x ) ) ( g `  x
) ) ) `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) >. } ) )
4020, 21, 22, 34, 39prdsvallem 14397 1  |-  ( ph  -> 
.<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   _Vcvv 2977    u. cun 3331    C_ wss 3333   {csn 3882   {cpr 3884   {ctp 3886   <.cop 3888   class class class wbr 4297   {copab 4354    e. cmpt 4355    X. cxp 4843   dom cdm 4845   ran crn 4846    o. ccom 4849   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   1stc1st 6580   2ndc2nd 6581   X_cixp 7268   supcsup 7695   0cc0 9287   RR*cxr 9422    < clt 9423   ndxcnx 14176   Basecbs 14179   +g cplusg 14243   .rcmulr 14244  Scalarcsca 14246   .scvsca 14247   .icip 14248  TopSetcts 14249   lecple 14250   distcds 14252   Hom chom 14254  compcco 14255   TopOpenctopn 14365   Xt_cpt 14382    gsumg cgsu 14384   X_scprds 14389
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-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-hom 14267  df-cco 14268  df-prds 14391
This theorem is referenced by:  prdsless  14406  prdsds  14407  prdstset  14409  prdshom  14410  prdsco  14411  prdsleval  14420  pwsle  14435
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