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Theorem broucube 31986
Description: Brouwer - or as Kulpa calls it, "Bohl-Brouwer" - fixed point theorem for the unit cube. Theorem on [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0  |-  ( ph  ->  N  e.  NN )
poimir.i  |-  I  =  ( ( 0 [,] 1 )  ^m  (
1 ... N ) )
poimir.r  |-  R  =  ( Xt_ `  (
( 1 ... N
)  X.  { (
topGen `  ran  (,) ) } ) )
broucube.1  |-  ( ph  ->  F  e.  ( ( Rt  I )  Cn  ( Rt  I ) ) )
Assertion
Ref Expression
broucube  |-  ( ph  ->  E. c  e.  I 
c  =  ( F `
 c ) )
Distinct variable groups:    ph, c    F, c    I, c    N, c    R, c

Proof of Theorem broucube
Dummy variables  n  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . 3  |-  ( ph  ->  N  e.  NN )
2 poimir.i . . 3  |-  I  =  ( ( 0 [,] 1 )  ^m  (
1 ... N ) )
3 poimir.r . . 3  |-  R  =  ( Xt_ `  (
( 1 ... N
)  X.  { (
topGen `  ran  (,) ) } ) )
4 elmapfn 7499 . . . . . . . 8  |-  ( x  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  x  Fn  ( 1 ... N
) )
54, 2eleq2s 2549 . . . . . . 7  |-  ( x  e.  I  ->  x  Fn  ( 1 ... N
) )
65adantl 468 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  Fn  ( 1 ... N
) )
7 broucube.1 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( Rt  I )  Cn  ( Rt  I ) ) )
8 ovex 6323 . . . . . . . . . . . . 13  |-  ( 1 ... N )  e. 
_V
9 retopon 21796 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
103pttoponconst 20624 . . . . . . . . . . . . 13  |-  ( ( ( 1 ... N
)  e.  _V  /\  ( topGen `  ran  (,) )  e.  (TopOn `  RR )
)  ->  R  e.  (TopOn `  ( RR  ^m  ( 1 ... N
) ) ) )
118, 9, 10mp2an 679 . . . . . . . . . . . 12  |-  R  e.  (TopOn `  ( RR  ^m  ( 1 ... N
) ) )
12 reex 9635 . . . . . . . . . . . . . 14  |-  RR  e.  _V
13 unitssre 11786 . . . . . . . . . . . . . 14  |-  ( 0 [,] 1 )  C_  RR
14 mapss 7519 . . . . . . . . . . . . . 14  |-  ( ( RR  e.  _V  /\  ( 0 [,] 1
)  C_  RR )  ->  ( ( 0 [,] 1 )  ^m  (
1 ... N ) ) 
C_  ( RR  ^m  ( 1 ... N
) ) )
1512, 13, 14mp2an 679 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 )  ^m  ( 1 ... N ) )  C_  ( RR  ^m  (
1 ... N ) )
162, 15eqsstri 3464 . . . . . . . . . . . 12  |-  I  C_  ( RR  ^m  (
1 ... N ) )
17 resttopon 20189 . . . . . . . . . . . 12  |-  ( ( R  e.  (TopOn `  ( RR  ^m  (
1 ... N ) ) )  /\  I  C_  ( RR  ^m  (
1 ... N ) ) )  ->  ( Rt  I
)  e.  (TopOn `  I ) )
1811, 16, 17mp2an 679 . . . . . . . . . . 11  |-  ( Rt  I )  e.  (TopOn `  I )
1918toponunii 19959 . . . . . . . . . 10  |-  I  = 
U. ( Rt  I )
2019, 19cnf 20274 . . . . . . . . 9  |-  ( F  e.  ( ( Rt  I )  Cn  ( Rt  I ) )  ->  F : I --> I )
217, 20syl 17 . . . . . . . 8  |-  ( ph  ->  F : I --> I )
2221ffvelrnda 6027 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  I )
23 elmapfn 7499 . . . . . . . 8  |-  ( ( F `  x )  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  ( F `  x )  Fn  ( 1 ... N
) )
2423, 2eleq2s 2549 . . . . . . 7  |-  ( ( F `  x )  e.  I  ->  ( F `  x )  Fn  ( 1 ... N
) )
2522, 24syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  Fn  ( 1 ... N
) )
268a1i 11 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
1 ... N )  e. 
_V )
27 inidm 3643 . . . . . 6  |-  ( ( 1 ... N )  i^i  ( 1 ... N ) )  =  ( 1 ... N
)
28 eqidd 2454 . . . . . 6  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
x `  n )  =  ( x `  n ) )
29 eqidd 2454 . . . . . 6  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  x
) `  n )  =  ( ( F `
 x ) `  n ) )
306, 25, 26, 26, 27, 28, 29offval 6543 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
x  oF  -  ( F `  x ) )  =  ( n  e.  ( 1 ... N )  |->  ( ( x `  n )  -  ( ( F `
 x ) `  n ) ) ) )
3130mpteq2dva 4492 . . . 4  |-  ( ph  ->  ( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) )  =  ( x  e.  I  |->  ( n  e.  ( 1 ... N
)  |->  ( ( x `
 n )  -  ( ( F `  x ) `  n
) ) ) ) )
3218a1i 11 . . . . 5  |-  ( ph  ->  ( Rt  I )  e.  (TopOn `  I ) )
338a1i 11 . . . . 5  |-  ( ph  ->  ( 1 ... N
)  e.  _V )
34 retop 21794 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  Top
3534fconst6 5778 . . . . . 6  |-  ( ( 1 ... N )  X.  { ( topGen ` 
ran  (,) ) } ) : ( 1 ... N ) --> Top
3635a1i 11 . . . . 5  |-  ( ph  ->  ( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) : ( 1 ... N ) --> Top )
3718a1i 11 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( Rt  I )  e.  (TopOn `  I ) )
38 eqid 2453 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3938cnfldtop 21816 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
40 cnrest2r 20315 . . . . . . . . . . 11  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) )  C_  ( ( Rt  I )  Cn  ( TopOpen ` fld ) ) )
4139, 40ax-mp 5 . . . . . . . . . 10  |-  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) )  C_  ( ( Rt  I )  Cn  ( TopOpen ` fld ) )
42 resmpt 5157 . . . . . . . . . . . . 13  |-  ( I 
C_  ( RR  ^m  ( 1 ... N
) )  ->  (
( x  e.  ( RR  ^m  ( 1 ... N ) ) 
|->  ( x `  n
) )  |`  I )  =  ( x  e.  I  |->  ( x `  n ) ) )
4316, 42ax-mp 5 . . . . . . . . . . . 12  |-  ( ( x  e.  ( RR 
^m  ( 1 ... N ) )  |->  ( x `  n ) )  |`  I )  =  ( x  e.  I  |->  ( x `  n ) )
4411toponunii 19959 . . . . . . . . . . . . . . 15  |-  ( RR 
^m  ( 1 ... N ) )  = 
U. R
4544, 3ptpjcn 20638 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... N
)  e.  _V  /\  ( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) : ( 1 ... N ) --> Top  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  ( RR 
^m  ( 1 ... N ) )  |->  ( x `  n ) )  e.  ( R  Cn  ( ( ( 1 ... N )  X.  { ( topGen ` 
ran  (,) ) } ) `
 n ) ) )
468, 35, 45mp3an12 1356 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... N )  ->  (
x  e.  ( RR 
^m  ( 1 ... N ) )  |->  ( x `  n ) )  e.  ( R  Cn  ( ( ( 1 ... N )  X.  { ( topGen ` 
ran  (,) ) } ) `
 n ) ) )
4744cnrest 20313 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ( RR  ^m  ( 1 ... N ) ) 
|->  ( x `  n
) )  e.  ( R  Cn  ( ( ( 1 ... N
)  X.  { (
topGen `  ran  (,) ) } ) `  n
) )  /\  I  C_  ( RR  ^m  (
1 ... N ) ) )  ->  ( (
x  e.  ( RR 
^m  ( 1 ... N ) )  |->  ( x `  n ) )  |`  I )  e.  ( ( Rt  I )  Cn  ( ( ( 1 ... N )  X.  { ( topGen ` 
ran  (,) ) } ) `
 n ) ) )
4846, 16, 47sylancl 669 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... N )  ->  (
( x  e.  ( RR  ^m  ( 1 ... N ) ) 
|->  ( x `  n
) )  |`  I )  e.  ( ( Rt  I )  Cn  ( ( ( 1 ... N
)  X.  { (
topGen `  ran  (,) ) } ) `  n
) ) )
4943, 48syl5eqelr 2536 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... N )  ->  (
x  e.  I  |->  ( x `  n ) )  e.  ( ( Rt  I )  Cn  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
) ) )
50 fvex 5880 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  e.  _V
5150fvconst2 6125 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
)  =  ( topGen ` 
ran  (,) ) )
5238tgioo2 21833 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
5351, 52syl6eq 2503 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
)  =  ( (
TopOpen ` fld )t  RR ) )
5453oveq2d 6311 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... N )  ->  (
( Rt  I )  Cn  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
) )  =  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) ) )
5549, 54eleqtrd 2533 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... N )  ->  (
x  e.  I  |->  ( x `  n ) )  e.  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) ) )
5641, 55sseldi 3432 . . . . . . . . 9  |-  ( n  e.  ( 1 ... N )  ->  (
x  e.  I  |->  ( x `  n ) )  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
5756adantl 468 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( x `  n ) )  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
5821feqmptd 5923 . . . . . . . . . . 11  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
5958, 7eqeltrrd 2532 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  I  |->  ( F `  x
) )  e.  ( ( Rt  I )  Cn  ( Rt  I ) ) )
6059adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( F `  x ) )  e.  ( ( Rt  I )  Cn  ( Rt  I ) ) )
61 fveq1 5869 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x `  n )  =  ( z `  n ) )
6261cbvmptv 4498 . . . . . . . . . 10  |-  ( x  e.  I  |->  ( x `
 n ) )  =  ( z  e.  I  |->  ( z `  n ) )
6362, 57syl5eqelr 2536 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
z  e.  I  |->  ( z `  n ) )  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
64 fveq1 5869 . . . . . . . . 9  |-  ( z  =  ( F `  x )  ->  (
z `  n )  =  ( ( F `
 x ) `  n ) )
6537, 60, 37, 63, 64cnmpt11 20690 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( ( F `  x
) `  n )
)  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
6638subcn 21910 . . . . . . . . 9  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
6766a1i 11 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) ) )
6837, 57, 65, 67cnmpt12f 20693 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( ( x `  n
)  -  ( ( F `  x ) `
 n ) ) )  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
69 elmapi 7498 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  x : ( 1 ... N ) --> ( 0 [,] 1 ) )
7069, 2eleq2s 2549 . . . . . . . . . . . . . 14  |-  ( x  e.  I  ->  x : ( 1 ... N ) --> ( 0 [,] 1 ) )
7170ffvelrnda 6027 . . . . . . . . . . . . 13  |-  ( ( x  e.  I  /\  n  e.  ( 1 ... N ) )  ->  ( x `  n )  e.  ( 0 [,] 1 ) )
7213, 71sseldi 3432 . . . . . . . . . . . 12  |-  ( ( x  e.  I  /\  n  e.  ( 1 ... N ) )  ->  ( x `  n )  e.  RR )
7372adantll 721 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
x `  n )  e.  RR )
74 elmapi 7498 . . . . . . . . . . . . . . 15  |-  ( ( F `  x )  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  ( F `  x ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
7574, 2eleq2s 2549 . . . . . . . . . . . . . 14  |-  ( ( F `  x )  e.  I  ->  ( F `  x ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
7622, 75syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
7776ffvelrnda 6027 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  x
) `  n )  e.  ( 0 [,] 1
) )
7813, 77sseldi 3432 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  x
) `  n )  e.  RR )
7973, 78resubcld 10054 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( x `  n
)  -  ( ( F `  x ) `
 n ) )  e.  RR )
8079an32s 814 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 1 ... N
) )  /\  x  e.  I )  ->  (
( x `  n
)  -  ( ( F `  x ) `
 n ) )  e.  RR )
81 eqid 2453 . . . . . . . . 9  |-  ( x  e.  I  |->  ( ( x `  n )  -  ( ( F `
 x ) `  n ) ) )  =  ( x  e.  I  |->  ( ( x `
 n )  -  ( ( F `  x ) `  n
) ) )
8280, 81fmptd 6051 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( ( x `  n
)  -  ( ( F `  x ) `
 n ) ) ) : I --> RR )
83 frn 5740 . . . . . . . 8  |-  ( ( x  e.  I  |->  ( ( x `  n
)  -  ( ( F `  x ) `
 n ) ) ) : I --> RR  ->  ran  ( x  e.  I  |->  ( ( x `  n )  -  (
( F `  x
) `  n )
) )  C_  RR )
8438cnfldtopon 21815 . . . . . . . . 9  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
85 ax-resscn 9601 . . . . . . . . 9  |-  RR  C_  CC
86 cnrest2 20314 . . . . . . . . 9  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  ran  ( x  e.  I  |->  ( ( x `  n )  -  (
( F `  x
) `  n )
) )  C_  RR  /\  RR  C_  CC )  ->  ( ( x  e.  I  |->  ( ( x `
 n )  -  ( ( F `  x ) `  n
) ) )  e.  ( ( Rt  I )  Cn  ( TopOpen ` fld ) )  <->  ( x  e.  I  |->  ( ( x `  n )  -  ( ( F `
 x ) `  n ) ) )  e.  ( ( Rt  I )  Cn  ( (
TopOpen ` fld )t  RR ) ) ) )
8784, 85, 86mp3an13 1357 . . . . . . . 8  |-  ( ran  ( x  e.  I  |->  ( ( x `  n )  -  (
( F `  x
) `  n )
) )  C_  RR  ->  ( ( x  e.  I  |->  ( ( x `
 n )  -  ( ( F `  x ) `  n
) ) )  e.  ( ( Rt  I )  Cn  ( TopOpen ` fld ) )  <->  ( x  e.  I  |->  ( ( x `  n )  -  ( ( F `
 x ) `  n ) ) )  e.  ( ( Rt  I )  Cn  ( (
TopOpen ` fld )t  RR ) ) ) )
8882, 83, 873syl 18 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( x  e.  I  |->  ( ( x `  n )  -  (
( F `  x
) `  n )
) )  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) )  <->  ( x  e.  I  |->  ( ( x `  n )  -  ( ( F `
 x ) `  n ) ) )  e.  ( ( Rt  I )  Cn  ( (
TopOpen ` fld )t  RR ) ) ) )
8968, 88mpbid 214 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( ( x `  n
)  -  ( ( F `  x ) `
 n ) ) )  e.  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) ) )
9054adantl 468 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( Rt  I )  Cn  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
) )  =  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) ) )
9189, 90eleqtrrd 2534 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( ( x `  n
)  -  ( ( F `  x ) `
 n ) ) )  e.  ( ( Rt  I )  Cn  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
) ) )
923, 32, 33, 36, 91ptcn 20654 . . . 4  |-  ( ph  ->  ( x  e.  I  |->  ( n  e.  ( 1 ... N ) 
|->  ( ( x `  n )  -  (
( F `  x
) `  n )
) ) )  e.  ( ( Rt  I )  Cn  R ) )
9331, 92eqeltrd 2531 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) )  e.  ( ( Rt  I )  Cn  R ) )
94 simpr2 1016 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  z  e.  I )
95 id 22 . . . . . . . . 9  |-  ( x  =  z  ->  x  =  z )
96 fveq2 5870 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
9795, 96oveq12d 6313 . . . . . . . 8  |-  ( x  =  z  ->  (
x  oF  -  ( F `  x ) )  =  ( z  oF  -  ( F `  z )
) )
98 eqid 2453 . . . . . . . 8  |-  ( x  e.  I  |->  ( x  oF  -  ( F `  x )
) )  =  ( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) )
99 ovex 6323 . . . . . . . 8  |-  ( z  oF  -  ( F `  z )
)  e.  _V
10097, 98, 99fvmpt 5953 . . . . . . 7  |-  ( z  e.  I  ->  (
( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) ) `
 z )  =  ( z  oF  -  ( F `  z ) ) )
101100fveq1d 5872 . . . . . 6  |-  ( z  e.  I  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  z ) `
 n )  =  ( ( z  oF  -  ( F `
 z ) ) `
 n ) )
10294, 101syl 17 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  z ) `
 n )  =  ( ( z  oF  -  ( F `
 z ) ) `
 n ) )
103 elmapfn 7499 . . . . . . . . . . . 12  |-  ( z  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  z  Fn  ( 1 ... N
) )
104103, 2eleq2s 2549 . . . . . . . . . . 11  |-  ( z  e.  I  ->  z  Fn  ( 1 ... N
) )
105104adantl 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z `  n )  =  0 )  /\  z  e.  I )  ->  z  Fn  ( 1 ... N ) )
10621ffvelrnda 6027 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  I )  ->  ( F `  z )  e.  I )
107 elmapfn 7499 . . . . . . . . . . . . 13  |-  ( ( F `  z )  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  ( F `  z )  Fn  ( 1 ... N
) )
108107, 2eleq2s 2549 . . . . . . . . . . . 12  |-  ( ( F `  z )  e.  I  ->  ( F `  z )  Fn  ( 1 ... N
) )
109106, 108syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  I )  ->  ( F `  z )  Fn  ( 1 ... N
) )
110109adantlr 722 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z `  n )  =  0 )  /\  z  e.  I )  ->  ( F `  z
)  Fn  ( 1 ... N ) )
1118a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z `  n )  =  0 )  /\  z  e.  I )  ->  ( 1 ... N
)  e.  _V )
112 simpllr 770 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( z `  n
)  =  0 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
z `  n )  =  0 )
113 eqidd 2454 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( z `  n
)  =  0 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  z
) `  n )  =  ( ( F `
 z ) `  n ) )
114105, 110, 111, 111, 27, 112, 113ofval 6545 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z `  n
)  =  0 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  ( 0  -  ( ( F `  z ) `
 n ) ) )
115 df-neg 9868 . . . . . . . . 9  |-  -u (
( F `  z
) `  n )  =  ( 0  -  ( ( F `  z ) `  n
) )
116114, 115syl6eqr 2505 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( z `  n
)  =  0 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  -u ( ( F `  z ) `  n
) )
117116exp41 615 . . . . . . 7  |-  ( ph  ->  ( ( z `  n )  =  0  ->  ( z  e.  I  ->  ( n  e.  ( 1 ... N
)  ->  ( (
z  oF  -  ( F `  z ) ) `  n )  =  -u ( ( F `
 z ) `  n ) ) ) ) )
118117com24 90 . . . . . 6  |-  ( ph  ->  ( n  e.  ( 1 ... N )  ->  ( z  e.  I  ->  ( (
z `  n )  =  0  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  -u ( ( F `  z ) `  n
) ) ) ) )
1191183imp2 1225 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  -u ( ( F `  z ) `  n
) )
120102, 119eqtrd 2487 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  z ) `
 n )  = 
-u ( ( F `
 z ) `  n ) )
121 elmapi 7498 . . . . . . . . . . . 12  |-  ( ( F `  z )  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  ( F `  z ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
122121, 2eleq2s 2549 . . . . . . . . . . 11  |-  ( ( F `  z )  e.  I  ->  ( F `  z ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
123106, 122syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  I )  ->  ( F `  z ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
124123ffvelrnda 6027 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  z
) `  n )  e.  ( 0 [,] 1
) )
125 0xr 9692 . . . . . . . . . 10  |-  0  e.  RR*
126 1re 9647 . . . . . . . . . . 11  |-  1  e.  RR
127126rexri 9698 . . . . . . . . . 10  |-  1  e.  RR*
128 iccgelb 11698 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  ( ( F `  z ) `
 n )  e.  ( 0 [,] 1
) )  ->  0  <_  ( ( F `  z ) `  n
) )
129125, 127, 128mp3an12 1356 . . . . . . . . 9  |-  ( ( ( F `  z
) `  n )  e.  ( 0 [,] 1
)  ->  0  <_  ( ( F `  z
) `  n )
)
130124, 129syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  0  <_  ( ( F `  z ) `  n
) )
13113, 124sseldi 3432 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  z
) `  n )  e.  RR )
132131le0neg2d 10193 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
0  <_  ( ( F `  z ) `  n )  <->  -u ( ( F `  z ) `
 n )  <_ 
0 ) )
133130, 132mpbid 214 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  -u (
( F `  z
) `  n )  <_  0 )
134133an32s 814 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( 1 ... N
) )  /\  z  e.  I )  ->  -u (
( F `  z
) `  n )  <_  0 )
135134anasss 653 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I ) )  ->  -u ( ( F `  z ) `  n
)  <_  0 )
1361353adantr3 1170 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  -u (
( F `  z
) `  n )  <_  0 )
137120, 136eqbrtrd 4426 . . 3  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  z ) `
 n )  <_ 
0 )
138 iccleub 11697 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  ( ( F `  z ) `
 n )  e.  ( 0 [,] 1
) )  ->  (
( F `  z
) `  n )  <_  1 )
139125, 127, 138mp3an12 1356 . . . . . . . . 9  |-  ( ( ( F `  z
) `  n )  e.  ( 0 [,] 1
)  ->  ( ( F `  z ) `  n )  <_  1
)
140124, 139syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  z
) `  n )  <_  1 )
141 1red 9663 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  1  e.  RR )
142141, 131subge0d 10210 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
0  <_  ( 1  -  ( ( F `
 z ) `  n ) )  <->  ( ( F `  z ) `  n )  <_  1
) )
143140, 142mpbird 236 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  0  <_  ( 1  -  (
( F `  z
) `  n )
) )
144143an32s 814 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( 1 ... N
) )  /\  z  e.  I )  ->  0  <_  ( 1  -  (
( F `  z
) `  n )
) )
145144anasss 653 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I ) )  -> 
0  <_  ( 1  -  ( ( F `
 z ) `  n ) ) )
1461453adantr3 1170 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  1 ) )  ->  0  <_  ( 1  -  (
( F `  z
) `  n )
) )
147 simpr2 1016 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  1 ) )  ->  z  e.  I )
148147, 101syl 17 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  1 ) )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  z ) `
 n )  =  ( ( z  oF  -  ( F `
 z ) ) `
 n ) )
149104adantl 468 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z `  n )  =  1 )  /\  z  e.  I )  ->  z  Fn  ( 1 ... N ) )
150109adantlr 722 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z `  n )  =  1 )  /\  z  e.  I )  ->  ( F `  z
)  Fn  ( 1 ... N ) )
1518a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z `  n )  =  1 )  /\  z  e.  I )  ->  ( 1 ... N
)  e.  _V )
152 simpllr 770 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z `  n
)  =  1 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
z `  n )  =  1 )
153 eqidd 2454 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z `  n
)  =  1 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  z
) `  n )  =  ( ( F `
 z ) `  n ) )
154149, 150, 151, 151, 27, 152, 153ofval 6545 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( z `  n
)  =  1 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  ( 1  -  ( ( F `  z ) `
 n ) ) )
155154exp41 615 . . . . . . 7  |-  ( ph  ->  ( ( z `  n )  =  1  ->  ( z  e.  I  ->  ( n  e.  ( 1 ... N
)  ->  ( (
z  oF  -  ( F `  z ) ) `  n )  =  ( 1  -  ( ( F `  z ) `  n
) ) ) ) ) )
156155com24 90 . . . . . 6  |-  ( ph  ->  ( n  e.  ( 1 ... N )  ->  ( z  e.  I  ->  ( (
z `  n )  =  1  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  ( 1  -  ( ( F `  z ) `
 n ) ) ) ) ) )
1571563imp2 1225 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  1 ) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  ( 1  -  ( ( F `  z ) `
 n ) ) )
158148, 157eqtrd 2487 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  1 ) )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  z ) `
 n )  =  ( 1  -  (
( F `  z
) `  n )
) )
159146, 158breqtrrd 4432 . . 3  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  1 ) )  ->  0  <_  ( ( ( x  e.  I  |->  ( x  oF  -  ( F `  x )
) ) `  z
) `  n )
)
1601, 2, 3, 93, 137, 159poimir 31985 . 2  |-  ( ph  ->  E. c  e.  I 
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  c )  =  ( ( 1 ... N )  X. 
{ 0 } ) )
161 id 22 . . . . . . . 8  |-  ( x  =  c  ->  x  =  c )
162 fveq2 5870 . . . . . . . 8  |-  ( x  =  c  ->  ( F `  x )  =  ( F `  c ) )
163161, 162oveq12d 6313 . . . . . . 7  |-  ( x  =  c  ->  (
x  oF  -  ( F `  x ) )  =  ( c  oF  -  ( F `  c )
) )
164 ovex 6323 . . . . . . 7  |-  ( c  oF  -  ( F `  c )
)  e.  _V
165163, 98, 164fvmpt 5953 . . . . . 6  |-  ( c  e.  I  ->  (
( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) ) `
 c )  =  ( c  oF  -  ( F `  c ) ) )
166165adantl 468 . . . . 5  |-  ( (
ph  /\  c  e.  I )  ->  (
( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) ) `
 c )  =  ( c  oF  -  ( F `  c ) ) )
167166eqeq1d 2455 . . . 4  |-  ( (
ph  /\  c  e.  I )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  c )  =  ( ( 1 ... N )  X. 
{ 0 } )  <-> 
( c  oF  -  ( F `  c ) )  =  ( ( 1 ... N )  X.  {
0 } ) ) )
168 elmapfn 7499 . . . . . . . . . . 11  |-  ( c  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  c  Fn  ( 1 ... N
) )
169168, 2eleq2s 2549 . . . . . . . . . 10  |-  ( c  e.  I  ->  c  Fn  ( 1 ... N
) )
170169adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  I )  ->  c  Fn  ( 1 ... N
) )
17121ffvelrnda 6027 . . . . . . . . . 10  |-  ( (
ph  /\  c  e.  I )  ->  ( F `  c )  e.  I )
172 elmapfn 7499 . . . . . . . . . . 11  |-  ( ( F `  c )  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  ( F `  c )  Fn  ( 1 ... N
) )
173172, 2eleq2s 2549 . . . . . . . . . 10  |-  ( ( F `  c )  e.  I  ->  ( F `  c )  Fn  ( 1 ... N
) )
174171, 173syl 17 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  I )  ->  ( F `  c )  Fn  ( 1 ... N
) )
1758a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  I )  ->  (
1 ... N )  e. 
_V )
176 eqidd 2454 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
c `  n )  =  ( c `  n ) )
177 eqidd 2454 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  c
) `  n )  =  ( ( F `
 c ) `  n ) )
178170, 174, 175, 175, 27, 176, 177ofval 6545 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( c  oF  -  ( F `  c ) ) `  n )  =  ( ( c `  n
)  -  ( ( F `  c ) `
 n ) ) )
179 c0ex 9642 . . . . . . . . . 10  |-  0  e.  _V
180179fvconst2 6125 . . . . . . . . 9  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
0 } ) `  n )  =  0 )
181180adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1 ... N )  X.  {
0 } ) `  n )  =  0 )
182178, 181eqeq12d 2468 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( c  oF  -  ( F `
 c ) ) `
 n )  =  ( ( ( 1 ... N )  X. 
{ 0 } ) `
 n )  <->  ( (
c `  n )  -  ( ( F `
 c ) `  n ) )  =  0 ) )
18313, 85sstri 3443 . . . . . . . . . 10  |-  ( 0 [,] 1 )  C_  CC
184 elmapi 7498 . . . . . . . . . . . 12  |-  ( c  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  c : ( 1 ... N ) --> ( 0 [,] 1 ) )
185184, 2eleq2s 2549 . . . . . . . . . . 11  |-  ( c  e.  I  ->  c : ( 1 ... N ) --> ( 0 [,] 1 ) )
186185ffvelrnda 6027 . . . . . . . . . 10  |-  ( ( c  e.  I  /\  n  e.  ( 1 ... N ) )  ->  ( c `  n )  e.  ( 0 [,] 1 ) )
187183, 186sseldi 3432 . . . . . . . . 9  |-  ( ( c  e.  I  /\  n  e.  ( 1 ... N ) )  ->  ( c `  n )  e.  CC )
188187adantll 721 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
c `  n )  e.  CC )
189 elmapi 7498 . . . . . . . . . . . 12  |-  ( ( F `  c )  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  ( F `  c ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
190189, 2eleq2s 2549 . . . . . . . . . . 11  |-  ( ( F `  c )  e.  I  ->  ( F `  c ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
191171, 190syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  c  e.  I )  ->  ( F `  c ) : ( 1 ... N ) --> ( 0 [,] 1 ) )
192191ffvelrnda 6027 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  c
) `  n )  e.  ( 0 [,] 1
) )
193183, 192sseldi 3432 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  c
) `  n )  e.  CC )
194188, 193subeq0ad 10001 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( c `  n )  -  (
( F `  c
) `  n )
)  =  0  <->  (
c `  n )  =  ( ( F `
 c ) `  n ) ) )
195182, 194bitrd 257 . . . . . 6  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( c  oF  -  ( F `
 c ) ) `
 n )  =  ( ( ( 1 ... N )  X. 
{ 0 } ) `
 n )  <->  ( c `  n )  =  ( ( F `  c
) `  n )
) )
196195ralbidva 2826 . . . . 5  |-  ( (
ph  /\  c  e.  I )  ->  ( A. n  e.  (
1 ... N ) ( ( c  oF  -  ( F `  c ) ) `  n )  =  ( ( ( 1 ... N )  X.  {
0 } ) `  n )  <->  A. n  e.  ( 1 ... N
) ( c `  n )  =  ( ( F `  c
) `  n )
) )
197170, 174, 175, 175, 27offn 6547 . . . . . 6  |-  ( (
ph  /\  c  e.  I )  ->  (
c  oF  -  ( F `  c ) )  Fn  ( 1 ... N ) )
198 fnconstg 5776 . . . . . . 7  |-  ( 0  e.  _V  ->  (
( 1 ... N
)  X.  { 0 } )  Fn  (
1 ... N ) )
199179, 198ax-mp 5 . . . . . 6  |-  ( ( 1 ... N )  X.  { 0 } )  Fn  ( 1 ... N )
200 eqfnfv 5981 . . . . . 6  |-  ( ( ( c  oF  -  ( F `  c ) )  Fn  ( 1 ... N
)  /\  ( (
1 ... N )  X. 
{ 0 } )  Fn  ( 1 ... N ) )  -> 
( ( c  oF  -  ( F `
 c ) )  =  ( ( 1 ... N )  X. 
{ 0 } )  <->  A. n  e.  (
1 ... N ) ( ( c  oF  -  ( F `  c ) ) `  n )  =  ( ( ( 1 ... N )  X.  {
0 } ) `  n ) ) )
201197, 199, 200sylancl 669 . . . . 5  |-  ( (
ph  /\  c  e.  I )  ->  (
( c  oF  -  ( F `  c ) )  =  ( ( 1 ... N )  X.  {
0 } )  <->  A. n  e.  ( 1 ... N
) ( ( c  oF  -  ( F `  c )
) `  n )  =  ( ( ( 1 ... N )  X.  { 0 } ) `  n ) ) )
202 eqfnfv 5981 . . . . . 6  |-  ( ( c  Fn  ( 1 ... N )  /\  ( F `  c )  Fn  ( 1 ... N ) )  -> 
( c  =  ( F `  c )  <->  A. n  e.  (
1 ... N ) ( c `  n )  =  ( ( F `
 c ) `  n ) ) )
203170, 174, 202syl2anc 667 . . . . 5  |-  ( (
ph  /\  c  e.  I )  ->  (
c  =  ( F `
 c )  <->  A. n  e.  ( 1 ... N
) ( c `  n )  =  ( ( F `  c
) `  n )
) )
204196, 201, 2033bitr4d 289 . . . 4  |-  ( (
ph  /\  c  e.  I )  ->  (
( c  oF  -  ( F `  c ) )  =  ( ( 1 ... N )  X.  {
0 } )  <->  c  =  ( F `  c ) ) )
205167, 204bitrd 257 . . 3  |-  ( (
ph  /\  c  e.  I )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  c )  =  ( ( 1 ... N )  X. 
{ 0 } )  <-> 
c  =  ( F `
 c ) ) )
206205rexbidva 2900 . 2  |-  ( ph  ->  ( E. c  e.  I  ( ( x  e.  I  |->  ( x  oF  -  ( F `  x )
) ) `  c
)  =  ( ( 1 ... N )  X.  { 0 } )  <->  E. c  e.  I 
c  =  ( F `
 c ) ) )
207160, 206mpbid 214 1  |-  ( ph  ->  E. c  e.  I 
c  =  ( F `
 c ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740   _Vcvv 3047    C_ wss 3406   {csn 3970   class class class wbr 4405    |-> cmpt 4464    X. cxp 4835   ran crn 4838    |` cres 4839    Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295    oFcof 6534    ^m cmap 7477   CCcc 9542   RRcr 9543   0cc0 9544   1c1 9545   RR*cxr 9679    <_ cle 9681    - cmin 9865   -ucneg 9866   NNcn 10616   (,)cioo 11642   [,]cicc 11645   ...cfz 11791   ↾t crest 15331   TopOpenctopn 15332   topGenctg 15348   Xt_cpt 15349  ℂfldccnfld 18982   Topctop 19929  TopOnctopon 19930    Cn ccn 20252    tX ctx 20587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-disj 4377  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  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-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-acn 8381  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-dvds 14318  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-ps 16458  df-tsr 16459  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-ntr 20047  df-cls 20048  df-lp 20164  df-cn 20255  df-cnp 20256  df-t1 20342  df-haus 20343  df-cmp 20414  df-tx 20589  df-hmeo 20782  df-hmph 20783  df-xms 21347  df-ms 21348  df-tms 21349  df-ii 21921
This theorem is referenced by: (None)
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