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Theorem broucube 31888
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 2530 . . . . . . 7  |-  ( x  e.  I  ->  x  Fn  ( 1 ... N
) )
65adantl 467 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  x  Fn  ( 1 ... N
) )
7 broucube.1 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( Rt  I )  Cn  ( Rt  I ) ) )
8 ovex 6330 . . . . . . . . . . . . 13  |-  ( 1 ... N )  e. 
_V
9 retopon 21771 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
103pttoponconst 20599 . . . . . . . . . . . . 13  |-  ( ( ( 1 ... N
)  e.  _V  /\  ( topGen `  ran  (,) )  e.  (TopOn `  RR )
)  ->  R  e.  (TopOn `  ( RR  ^m  ( 1 ... N
) ) ) )
118, 9, 10mp2an 676 . . . . . . . . . . . 12  |-  R  e.  (TopOn `  ( RR  ^m  ( 1 ... N
) ) )
12 reex 9631 . . . . . . . . . . . . . 14  |-  RR  e.  _V
13 unitssre 11780 . . . . . . . . . . . . . 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 676 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 )  ^m  ( 1 ... N ) )  C_  ( RR  ^m  (
1 ... N ) )
162, 15eqsstri 3494 . . . . . . . . . . . 12  |-  I  C_  ( RR  ^m  (
1 ... N ) )
17 resttopon 20164 . . . . . . . . . . . 12  |-  ( ( R  e.  (TopOn `  ( RR  ^m  (
1 ... N ) ) )  /\  I  C_  ( RR  ^m  (
1 ... N ) ) )  ->  ( Rt  I
)  e.  (TopOn `  I ) )
1811, 16, 17mp2an 676 . . . . . . . . . . 11  |-  ( Rt  I )  e.  (TopOn `  I )
1918toponunii 19934 . . . . . . . . . 10  |-  I  = 
U. ( Rt  I )
2019, 19cnf 20249 . . . . . . . . 9  |-  ( F  e.  ( ( Rt  I )  Cn  ( Rt  I ) )  ->  F : I --> I )
217, 20syl 17 . . . . . . . 8  |-  ( ph  ->  F : I --> I )
2221ffvelrnda 6034 . . . . . . 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 2530 . . . . . . 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 3671 . . . . . 6  |-  ( ( 1 ... N )  i^i  ( 1 ... N ) )  =  ( 1 ... N
)
28 eqidd 2423 . . . . . 6  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
x `  n )  =  ( x `  n ) )
29 eqidd 2423 . . . . . 6  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  x
) `  n )  =  ( ( F `
 x ) `  n ) )
306, 25, 26, 26, 27, 28, 29offval 6549 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
x  oF  -  ( F `  x ) )  =  ( n  e.  ( 1 ... N )  |->  ( ( x `  n )  -  ( ( F `
 x ) `  n ) ) ) )
3130mpteq2dva 4507 . . . 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 21769 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  Top
3534fconst6 5787 . . . . . 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 2422 . . . . . . . . . . . 12  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3938cnfldtop 21791 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  e.  Top
40 cnrest2r 20290 . . . . . . . . . . 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 5170 . . . . . . . . . . . . 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 19934 . . . . . . . . . . . . . . 15  |-  ( RR 
^m  ( 1 ... N ) )  = 
U. R
4544, 3ptpjcn 20613 . . . . . . . . . . . . . 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 1350 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... N )  ->  (
x  e.  ( RR 
^m  ( 1 ... N ) )  |->  ( x `  n ) )  e.  ( R  Cn  ( ( ( 1 ... N )  X.  { ( topGen ` 
ran  (,) ) } ) `
 n ) ) )
4744cnrest 20288 . . . . . . . . . . . . 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 666 . . . . . . . . . . . 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 2515 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... N )  ->  (
x  e.  I  |->  ( x `  n ) )  e.  ( ( Rt  I )  Cn  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
) ) )
50 fvex 5888 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  e.  _V
5150fvconst2 6132 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
)  =  ( topGen ` 
ran  (,) ) )
5238tgioo2 21808 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
5351, 52syl6eq 2479 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
)  =  ( (
TopOpen ` fld )t  RR ) )
5453oveq2d 6318 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... N )  ->  (
( Rt  I )  Cn  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
) )  =  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) ) )
5549, 54eleqtrd 2512 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... N )  ->  (
x  e.  I  |->  ( x `  n ) )  e.  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) ) )
5641, 55sseldi 3462 . . . . . . . . 9  |-  ( n  e.  ( 1 ... N )  ->  (
x  e.  I  |->  ( x `  n ) )  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
5756adantl 467 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( x `  n ) )  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
5821feqmptd 5931 . . . . . . . . . . 11  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
5958, 7eqeltrrd 2511 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  I  |->  ( F `  x
) )  e.  ( ( Rt  I )  Cn  ( Rt  I ) ) )
6059adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( F `  x ) )  e.  ( ( Rt  I )  Cn  ( Rt  I ) ) )
61 fveq1 5877 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x `  n )  =  ( z `  n ) )
6261cbvmptv 4513 . . . . . . . . . 10  |-  ( x  e.  I  |->  ( x `
 n ) )  =  ( z  e.  I  |->  ( z `  n ) )
6362, 57syl5eqelr 2515 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
z  e.  I  |->  ( z `  n ) )  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
64 fveq1 5877 . . . . . . . . 9  |-  ( z  =  ( F `  x )  ->  (
z `  n )  =  ( ( F `
 x ) `  n ) )
6537, 60, 37, 63, 64cnmpt11 20665 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( ( F `  x
) `  n )
)  e.  ( ( Rt  I )  Cn  ( TopOpen
` fld
) ) )
6638subcn 21885 . . . . . . . . 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 20668 . . . . . . 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 2530 . . . . . . . . . . . . . 14  |-  ( x  e.  I  ->  x : ( 1 ... N ) --> ( 0 [,] 1 ) )
7170ffvelrnda 6034 . . . . . . . . . . . . 13  |-  ( ( x  e.  I  /\  n  e.  ( 1 ... N ) )  ->  ( x `  n )  e.  ( 0 [,] 1 ) )
7213, 71sseldi 3462 . . . . . . . . . . . 12  |-  ( ( x  e.  I  /\  n  e.  ( 1 ... N ) )  ->  ( x `  n )  e.  RR )
7372adantll 718 . . . . . . . . . . 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 2530 . . . . . . . . . . . . . 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 6034 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  x
) `  n )  e.  ( 0 [,] 1
) )
7813, 77sseldi 3462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  x
) `  n )  e.  RR )
7973, 78resubcld 10048 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( x `  n
)  -  ( ( F `  x ) `
 n ) )  e.  RR )
8079an32s 811 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( 1 ... N
) )  /\  x  e.  I )  ->  (
( x `  n
)  -  ( ( F `  x ) `
 n ) )  e.  RR )
81 eqid 2422 . . . . . . . . 9  |-  ( x  e.  I  |->  ( ( x `  n )  -  ( ( F `
 x ) `  n ) ) )  =  ( x  e.  I  |->  ( ( x `
 n )  -  ( ( F `  x ) `  n
) ) )
8280, 81fmptd 6058 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( ( x `  n
)  -  ( ( F `  x ) `
 n ) ) ) : I --> RR )
83 frn 5749 . . . . . . . 8  |-  ( ( x  e.  I  |->  ( ( x `  n
)  -  ( ( F `  x ) `
 n ) ) ) : I --> RR  ->  ran  ( x  e.  I  |->  ( ( x `  n )  -  (
( F `  x
) `  n )
) )  C_  RR )
8438cnfldtopon 21790 . . . . . . . . 9  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
85 ax-resscn 9597 . . . . . . . . 9  |-  RR  C_  CC
86 cnrest2 20289 . . . . . . . . 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 1351 . . . . . . . 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 213 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
x  e.  I  |->  ( ( x `  n
)  -  ( ( F `  x ) `
 n ) ) )  e.  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) ) )
9054adantl 467 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( Rt  I )  Cn  (
( ( 1 ... N )  X.  {
( topGen `  ran  (,) ) } ) `  n
) )  =  ( ( Rt  I )  Cn  (
( TopOpen ` fld )t  RR ) ) )
9189, 90eleqtrrd 2513 . . . . 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 20629 . . . 4  |-  ( ph  ->  ( x  e.  I  |->  ( n  e.  ( 1 ... N ) 
|->  ( ( x `  n )  -  (
( F `  x
) `  n )
) ) )  e.  ( ( Rt  I )  Cn  R ) )
9331, 92eqeltrd 2510 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) )  e.  ( ( Rt  I )  Cn  R ) )
94 simpr2 1012 . . . . . 6  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  z  e.  I )
95 id 23 . . . . . . . . 9  |-  ( x  =  z  ->  x  =  z )
96 fveq2 5878 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
9795, 96oveq12d 6320 . . . . . . . 8  |-  ( x  =  z  ->  (
x  oF  -  ( F `  x ) )  =  ( z  oF  -  ( F `  z )
) )
98 eqid 2422 . . . . . . . 8  |-  ( x  e.  I  |->  ( x  oF  -  ( F `  x )
) )  =  ( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) )
99 ovex 6330 . . . . . . . 8  |-  ( z  oF  -  ( F `  z )
)  e.  _V
10097, 98, 99fvmpt 5961 . . . . . . 7  |-  ( z  e.  I  ->  (
( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) ) `
 z )  =  ( z  oF  -  ( F `  z ) ) )
101100fveq1d 5880 . . . . . 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 2530 . . . . . . . . . . 11  |-  ( z  e.  I  ->  z  Fn  ( 1 ... N
) )
105104adantl 467 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z `  n )  =  0 )  /\  z  e.  I )  ->  z  Fn  ( 1 ... N ) )
10621ffvelrnda 6034 . . . . . . . . . . . 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 2530 . . . . . . . . . . . 12  |-  ( ( F `  z )  e.  I  ->  ( F `  z )  Fn  ( 1 ... N
) )
109106, 108syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  I )  ->  ( F `  z )  Fn  ( 1 ... N
) )
110109adantlr 719 . . . . . . . . . 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 767 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( z `  n
)  =  0 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
z `  n )  =  0 )
113 eqidd 2423 . . . . . . . . . 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 6551 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z `  n
)  =  0 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  ( 0  -  ( ( F `  z ) `
 n ) ) )
115 df-neg 9864 . . . . . . . . 9  |-  -u (
( F `  z
) `  n )  =  ( 0  -  ( ( F `  z ) `  n
) )
116114, 115syl6eqr 2481 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( z `  n
)  =  0 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  -u ( ( F `  z ) `  n
) )
117116exp41 613 . . . . . . 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 1220 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  -u ( ( F `  z ) `  n
) )
120102, 119eqtrd 2463 . . . 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 2530 . . . . . . . . . . 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 6034 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  z
) `  n )  e.  ( 0 [,] 1
) )
125 0xr 9688 . . . . . . . . . 10  |-  0  e.  RR*
126 1re 9643 . . . . . . . . . . 11  |-  1  e.  RR
127126rexri 9694 . . . . . . . . . 10  |-  1  e.  RR*
128 iccgelb 11692 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  ( ( F `  z ) `
 n )  e.  ( 0 [,] 1
) )  ->  0  <_  ( ( F `  z ) `  n
) )
129125, 127, 128mp3an12 1350 . . . . . . . . 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 3462 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  z
) `  n )  e.  RR )
132131le0neg2d 10187 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
0  <_  ( ( F `  z ) `  n )  <->  -u ( ( F `  z ) `
 n )  <_ 
0 ) )
133130, 132mpbid 213 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  -u (
( F `  z
) `  n )  <_  0 )
134133an32s 811 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( 1 ... N
) )  /\  z  e.  I )  ->  -u (
( F `  z
) `  n )  <_  0 )
135134anasss 651 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I ) )  ->  -u ( ( F `  z ) `  n
)  <_  0 )
1361353adantr3 1166 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  -u (
( F `  z
) `  n )  <_  0 )
137120, 136eqbrtrd 4441 . . 3  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  0 ) )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  z ) `
 n )  <_ 
0 )
138 iccleub 11691 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  1  e.  RR*  /\  ( ( F `  z ) `
 n )  e.  ( 0 [,] 1
) )  ->  (
( F `  z
) `  n )  <_  1 )
139125, 127, 138mp3an12 1350 . . . . . . . . 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 9659 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  1  e.  RR )
142141, 131subge0d 10204 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
0  <_  ( 1  -  ( ( F `
 z ) `  n ) )  <->  ( ( F `  z ) `  n )  <_  1
) )
143140, 142mpbird 235 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  0  <_  ( 1  -  (
( F `  z
) `  n )
) )
144143an32s 811 . . . . . 6  |-  ( ( ( ph  /\  n  e.  ( 1 ... N
) )  /\  z  e.  I )  ->  0  <_  ( 1  -  (
( F `  z
) `  n )
) )
145144anasss 651 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I ) )  -> 
0  <_  ( 1  -  ( ( F `
 z ) `  n ) ) )
1461453adantr3 1166 . . . 4  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  1 ) )  ->  0  <_  ( 1  -  (
( F `  z
) `  n )
) )
147 simpr2 1012 . . . . . 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 467 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z `  n )  =  1 )  /\  z  e.  I )  ->  z  Fn  ( 1 ... N ) )
150109adantlr 719 . . . . . . . . 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 767 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z `  n
)  =  1 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
z `  n )  =  1 )
153 eqidd 2423 . . . . . . . . 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 6551 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( z `  n
)  =  1 )  /\  z  e.  I
)  /\  n  e.  ( 1 ... N
) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  ( 1  -  ( ( F `  z ) `
 n ) ) )
155154exp41 613 . . . . . . 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 1220 . . . . 5  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  z  e.  I  /\  ( z `  n )  =  1 ) )  ->  (
( z  oF  -  ( F `  z ) ) `  n )  =  ( 1  -  ( ( F `  z ) `
 n ) ) )
158148, 157eqtrd 2463 . . . 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 4447 . . 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 31887 . 2  |-  ( ph  ->  E. c  e.  I 
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  c )  =  ( ( 1 ... N )  X. 
{ 0 } ) )
161 id 23 . . . . . . . 8  |-  ( x  =  c  ->  x  =  c )
162 fveq2 5878 . . . . . . . 8  |-  ( x  =  c  ->  ( F `  x )  =  ( F `  c ) )
163161, 162oveq12d 6320 . . . . . . 7  |-  ( x  =  c  ->  (
x  oF  -  ( F `  x ) )  =  ( c  oF  -  ( F `  c )
) )
164 ovex 6330 . . . . . . 7  |-  ( c  oF  -  ( F `  c )
)  e.  _V
165163, 98, 164fvmpt 5961 . . . . . 6  |-  ( c  e.  I  ->  (
( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) ) `
 c )  =  ( c  oF  -  ( F `  c ) ) )
166165adantl 467 . . . . 5  |-  ( (
ph  /\  c  e.  I )  ->  (
( x  e.  I  |->  ( x  oF  -  ( F `  x ) ) ) `
 c )  =  ( c  oF  -  ( F `  c ) ) )
167166eqeq1d 2424 . . . 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 2530 . . . . . . . . . 10  |-  ( c  e.  I  ->  c  Fn  ( 1 ... N
) )
170169adantl 467 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  I )  ->  c  Fn  ( 1 ... N
) )
17121ffvelrnda 6034 . . . . . . . . . 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 2530 . . . . . . . . . 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 2423 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
c `  n )  =  ( c `  n ) )
177 eqidd 2423 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  c
) `  n )  =  ( ( F `
 c ) `  n ) )
178170, 174, 175, 175, 27, 176, 177ofval 6551 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( c  oF  -  ( F `  c ) ) `  n )  =  ( ( c `  n
)  -  ( ( F `  c ) `
 n ) ) )
179 c0ex 9638 . . . . . . . . . 10  |-  0  e.  _V
180179fvconst2 6132 . . . . . . . . 9  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
0 } ) `  n )  =  0 )
181180adantl 467 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1 ... N )  X.  {
0 } ) `  n )  =  0 )
182178, 181eqeq12d 2444 . . . . . . 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 3473 . . . . . . . . . 10  |-  ( 0 [,] 1 )  C_  CC
184 elmapi 7498 . . . . . . . . . . . 12  |-  ( c  e.  ( ( 0 [,] 1 )  ^m  ( 1 ... N
) )  ->  c : ( 1 ... N ) --> ( 0 [,] 1 ) )
185184, 2eleq2s 2530 . . . . . . . . . . 11  |-  ( c  e.  I  ->  c : ( 1 ... N ) --> ( 0 [,] 1 ) )
186185ffvelrnda 6034 . . . . . . . . . 10  |-  ( ( c  e.  I  /\  n  e.  ( 1 ... N ) )  ->  ( c `  n )  e.  ( 0 [,] 1 ) )
187183, 186sseldi 3462 . . . . . . . . 9  |-  ( ( c  e.  I  /\  n  e.  ( 1 ... N ) )  ->  ( c `  n )  e.  CC )
188187adantll 718 . . . . . . . 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 2530 . . . . . . . . . . 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 6034 . . . . . . . . 9  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  c
) `  n )  e.  ( 0 [,] 1
) )
193183, 192sseldi 3462 . . . . . . . 8  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  c
) `  n )  e.  CC )
194188, 193subeq0ad 9997 . . . . . . 7  |-  ( ( ( ph  /\  c  e.  I )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( c `  n )  -  (
( F `  c
) `  n )
)  =  0  <->  (
c `  n )  =  ( ( F `
 c ) `  n ) ) )
195182, 194bitrd 256 . . . . . 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 2861 . . . . 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 6553 . . . . . 6  |-  ( (
ph  /\  c  e.  I )  ->  (
c  oF  -  ( F `  c ) )  Fn  ( 1 ... N ) )
198 fnconstg 5785 . . . . . . 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 5988 . . . . . 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 666 . . . . 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 5988 . . . . . 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 665 . . . . 5  |-  ( (
ph  /\  c  e.  I )  ->  (
c  =  ( F `
 c )  <->  A. n  e.  ( 1 ... N
) ( c `  n )  =  ( ( F `  c
) `  n )
) )
204196, 201, 2033bitr4d 288 . . . 4  |-  ( (
ph  /\  c  e.  I )  ->  (
( c  oF  -  ( F `  c ) )  =  ( ( 1 ... N )  X.  {
0 } )  <->  c  =  ( F `  c ) ) )
205167, 204bitrd 256 . . 3  |-  ( (
ph  /\  c  e.  I )  ->  (
( ( x  e.  I  |->  ( x  oF  -  ( F `
 x ) ) ) `  c )  =  ( ( 1 ... N )  X. 
{ 0 } )  <-> 
c  =  ( F `
 c ) ) )
206205rexbidva 2936 . 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 213 1  |-  ( ph  ->  E. c  e.  I 
c  =  ( F `
 c ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   _Vcvv 3081    C_ wss 3436   {csn 3996   class class class wbr 4420    |-> cmpt 4479    X. cxp 4848   ran crn 4851    |` cres 4852    Fn wfn 5593   -->wf 5594   ` cfv 5598  (class class class)co 6302    oFcof 6540    ^m cmap 7477   CCcc 9538   RRcr 9539   0cc0 9540   1c1 9541   RR*cxr 9675    <_ cle 9677    - cmin 9861   -ucneg 9862   NNcn 10610   (,)cioo 11636   [,]cicc 11639   ...cfz 11785   ↾t crest 15307   TopOpenctopn 15308   topGenctg 15324   Xt_cpt 15325  ℂfldccnfld 18958   Topctop 19904  TopOnctopon 19905    Cn ccn 20227    tX ctx 20562
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-disj 4392  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  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 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-acn 8378  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-sum 13741  df-dvds 14294  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-ps 16434  df-tsr 16435  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-lp 20139  df-cn 20230  df-cnp 20231  df-t1 20317  df-haus 20318  df-cmp 20389  df-tx 20564  df-hmeo 20757  df-hmph 20758  df-xms 21322  df-ms 21323  df-tms 21324  df-ii 21896
This theorem is referenced by: (None)
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