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Theorem cvxscon 24883
Description: A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
Hypotheses
Ref Expression
cvxpcon.1  |-  ( ph  ->  S  C_  CC )
cvxpcon.2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( ( t  x.  x )  +  ( ( 1  -  t
)  x.  y ) )  e.  S )
cvxpcon.3  |-  J  =  ( TopOpen ` fld )
cvxpcon.4  |-  K  =  ( Jt  S )
Assertion
Ref Expression
cvxscon  |-  ( ph  ->  K  e. SCon )
Distinct variable groups:    t, J    x, t, y, K    ph, t, x, y    t, S, x, y
Allowed substitution hints:    J( x, y)

Proof of Theorem cvxscon
Dummy variables  z 
f  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvxpcon.1 . . 3  |-  ( ph  ->  S  C_  CC )
2 cvxpcon.2 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( ( t  x.  x )  +  ( ( 1  -  t
)  x.  y ) )  e.  S )
3 cvxpcon.3 . . 3  |-  J  =  ( TopOpen ` fld )
4 cvxpcon.4 . . 3  |-  K  =  ( Jt  S )
51, 2, 3, 4cvxpcon 24882 . 2  |-  ( ph  ->  K  e. PCon )
6 simprl 733 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f  e.  ( II 
Cn  K ) )
7 pcontop 24865 . . . . . . . . . 10  |-  ( K  e. PCon  ->  K  e.  Top )
85, 7syl 16 . . . . . . . . 9  |-  ( ph  ->  K  e.  Top )
98adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  K  e.  Top )
10 eqid 2404 . . . . . . . . 9  |-  U. K  =  U. K
1110toptopon 16953 . . . . . . . 8  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
129, 11sylib 189 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  K  e.  (TopOn `  U. K ) )
13 iiuni 18864 . . . . . . . . . 10  |-  ( 0 [,] 1 )  = 
U. II
1413, 10cnf 17264 . . . . . . . . 9  |-  ( f  e.  ( II  Cn  K )  ->  f : ( 0 [,] 1 ) --> U. K
)
156, 14syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f : ( 0 [,] 1 ) --> U. K )
16 0elunit 10971 . . . . . . . 8  |-  0  e.  ( 0 [,] 1
)
17 ffvelrn 5827 . . . . . . . 8  |-  ( ( f : ( 0 [,] 1 ) --> U. K  /\  0  e.  ( 0 [,] 1
) )  ->  (
f `  0 )  e.  U. K )
1815, 16, 17sylancl 644 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f `  0
)  e.  U. K
)
19 eqid 2404 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )
2019pcoptcl 18999 . . . . . . 7  |-  ( ( K  e.  (TopOn `  U. K )  /\  (
f `  0 )  e.  U. K )  -> 
( ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  e.  ( II  Cn  K )  /\  (
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) ` 
0 )  =  ( f `  0 )  /\  ( ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) `  1 )  =  ( f ` 
0 ) ) )
2112, 18, 20syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } )  e.  ( II  Cn  K )  /\  (
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) ` 
0 )  =  ( f `  0 )  /\  ( ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) `  1 )  =  ( f ` 
0 ) ) )
2221simp1d 969 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( ( 0 [,] 1 )  X.  {
( f `  0
) } )  e.  ( II  Cn  K
) )
23 iitopon 18862 . . . . . . . . . . 11  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
2423a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  II  e.  (TopOn `  (
0 [,] 1 ) ) )
253dfii3 18866 . . . . . . . . . . . 12  |-  II  =  ( Jt  ( 0 [,] 1 ) )
263cnfldtopon 18770 . . . . . . . . . . . . 13  |-  J  e.  (TopOn `  CC )
2726a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  J  e.  (TopOn `  CC ) )
28 unitssre 10998 . . . . . . . . . . . . . 14  |-  ( 0 [,] 1 )  C_  RR
29 ax-resscn 9003 . . . . . . . . . . . . . 14  |-  RR  C_  CC
3028, 29sstri 3317 . . . . . . . . . . . . 13  |-  ( 0 [,] 1 )  C_  CC
3130a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( 0 [,] 1
)  C_  CC )
3227, 27cnmpt2nd 17654 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  CC ,  t  e.  CC  |->  t )  e.  ( ( J  tX  J
)  Cn  J ) )
3325, 27, 31, 25, 27, 31, 32cnmpt2res 17662 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  t )  e.  ( ( II  tX  II )  Cn  J ) )
341adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  S  C_  CC )
35 resttopon 17179 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  CC )  /\  S  C_  CC )  ->  ( Jt  S )  e.  (TopOn `  S ) )
3626, 1, 35sylancr 645 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( Jt  S )  e.  (TopOn `  S ) )
374, 36syl5eqel 2488 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  K  e.  (TopOn `  S ) )
38 toponuni 16947 . . . . . . . . . . . . . . . 16  |-  ( K  e.  (TopOn `  S
)  ->  S  =  U. K )
3937, 38syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S  =  U. K
)
4039adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  S  =  U. K )
4118, 40eleqtrrd 2481 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f `  0
)  e.  S )
4234, 41sseldd 3309 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f `  0
)  e.  CC )
4324, 24, 27, 42cnmpt2c 17655 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( f `  0
) )  e.  ( ( II  tX  II )  Cn  J ) )
443mulcn 18850 . . . . . . . . . . . 12  |-  x.  e.  ( ( J  tX  J )  Cn  J
)
4544a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  x.  e.  ( ( J 
tX  J )  Cn  J ) )
4624, 24, 33, 43, 45cnmpt22f 17660 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( t  x.  (
f `  0 )
) )  e.  ( ( II  tX  II )  Cn  J ) )
47 ax-1cn 9004 . . . . . . . . . . . . . . 15  |-  1  e.  CC
4847a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
1  e.  CC )
4927, 27, 27, 48cnmpt2c 17655 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  CC ,  t  e.  CC  |->  1 )  e.  ( ( J  tX  J
)  Cn  J ) )
503subcn 18849 . . . . . . . . . . . . . 14  |-  -  e.  ( ( J  tX  J )  Cn  J
)
5150a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  -  e.  ( ( J  tX  J )  Cn  J ) )
5227, 27, 49, 32, 51cnmpt22f 17660 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  CC ,  t  e.  CC  |->  ( 1  -  t
) )  e.  ( ( J  tX  J
)  Cn  J ) )
5325, 27, 31, 25, 27, 31, 52cnmpt2res 17662 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( 1  -  t
) )  e.  ( ( II  tX  II )  Cn  J ) )
5424, 24cnmpt1st 17653 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  z )  e.  ( ( II  tX  II )  Cn  II ) )
553cnfldtop 18771 . . . . . . . . . . . . . 14  |-  J  e. 
Top
56 cnrest2r 17305 . . . . . . . . . . . . . 14  |-  ( J  e.  Top  ->  (
II  Cn  ( Jt  S
) )  C_  (
II  Cn  J )
)
5755, 56ax-mp 8 . . . . . . . . . . . . 13  |-  ( II 
Cn  ( Jt  S ) )  C_  ( II  Cn  J )
584oveq2i 6051 . . . . . . . . . . . . . 14  |-  ( II 
Cn  K )  =  ( II  Cn  ( Jt  S ) )
596, 58syl6eleq 2494 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f  e.  ( II 
Cn  ( Jt  S ) ) )
6057, 59sseldi 3306 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f  e.  ( II 
Cn  J ) )
6124, 24, 54, 60cnmpt21f 17657 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( f `  z
) )  e.  ( ( II  tX  II )  Cn  J ) )
6224, 24, 53, 61, 45cnmpt22f 17660 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( ( 1  -  t )  x.  (
f `  z )
) )  e.  ( ( II  tX  II )  Cn  J ) )
633addcn 18848 . . . . . . . . . . 11  |-  +  e.  ( ( J  tX  J )  Cn  J
)
6463a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  +  e.  ( ( J  tX  J )  Cn  J ) )
6524, 24, 46, 62, 64cnmpt22f 17660 . . . . . . . . 9  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  ( f `
 z ) ) ) )  e.  ( ( II  tX  II )  Cn  J ) )
6641adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( f `  0
)  e.  S )
6715adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
f : ( 0 [,] 1 ) --> U. K )
68 simprl 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
z  e.  ( 0 [,] 1 ) )
6967, 68ffvelrnd 5830 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( f `  z
)  e.  U. K
)
7040adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  ->  S  =  U. K )
7169, 70eleqtrrd 2481 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( f `  z
)  e.  S )
7223exp2 1171 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( x  e.  S  ->  ( y  e.  S  ->  ( t  e.  ( 0 [,] 1 )  ->  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  e.  S
) ) ) )
7372imp42 578 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  t  e.  ( 0 [,] 1
) )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  e.  S )
7473an32s 780 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  t  e.  ( 0 [,] 1
) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( (
t  x.  x )  +  ( ( 1  -  t )  x.  y ) )  e.  S )
7574ralrimivva 2758 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  t  e.  ( 0 [,] 1
) )  ->  A. x  e.  S  A. y  e.  S  ( (
t  x.  x )  +  ( ( 1  -  t )  x.  y ) )  e.  S )
7675ad2ant2rl 730 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  ->  A. x  e.  S  A. y  e.  S  ( ( t  x.  x )  +  ( ( 1  -  t
)  x.  y ) )  e.  S )
77 oveq2 6048 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( f ` 
0 )  ->  (
t  x.  x )  =  ( t  x.  ( f `  0
) ) )
7877oveq1d 6055 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( f ` 
0 )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  =  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  y
) ) )
7978eleq1d 2470 . . . . . . . . . . . . . . 15  |-  ( x  =  ( f ` 
0 )  ->  (
( ( t  x.  x )  +  ( ( 1  -  t
)  x.  y ) )  e.  S  <->  ( (
t  x.  ( f `
 0 ) )  +  ( ( 1  -  t )  x.  y ) )  e.  S ) )
80 oveq2 6048 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( f `  z )  ->  (
( 1  -  t
)  x.  y )  =  ( ( 1  -  t )  x.  ( f `  z
) ) )
8180oveq2d 6056 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( f `  z )  ->  (
( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  y ) )  =  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )
8281eleq1d 2470 . . . . . . . . . . . . . . 15  |-  ( y  =  ( f `  z )  ->  (
( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  y ) )  e.  S  <->  ( (
t  x.  ( f `
 0 ) )  +  ( ( 1  -  t )  x.  ( f `  z
) ) )  e.  S ) )
8379, 82rspc2va 3019 . . . . . . . . . . . . . 14  |-  ( ( ( ( f ` 
0 )  e.  S  /\  ( f `  z
)  e.  S )  /\  A. x  e.  S  A. y  e.  S  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  e.  S
)  ->  ( (
t  x.  ( f `
 0 ) )  +  ( ( 1  -  t )  x.  ( f `  z
) ) )  e.  S )
8466, 71, 76, 83syl21anc 1183 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  ( f `
 z ) ) )  e.  S )
8584ralrimivva 2758 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  A. z  e.  (
0 [,] 1 ) A. t  e.  ( 0 [,] 1 ) ( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  ( f `
 z ) ) )  e.  S )
86 eqid 2404 . . . . . . . . . . . . 13  |-  ( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( ( t  x.  ( f `
 0 ) )  +  ( ( 1  -  t )  x.  ( f `  z
) ) ) )  =  ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )
8786fmpt2 6377 . . . . . . . . . . . 12  |-  ( A. z  e.  ( 0 [,] 1 ) A. t  e.  ( 0 [,] 1 ) ( ( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  ( f `  z ) ) )  e.  S  <->  ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) : ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) --> S )
8885, 87sylib 189 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  ( f `
 z ) ) ) ) : ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) --> S )
89 frn 5556 . . . . . . . . . . 11  |-  ( ( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( ( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  ( f `  z ) ) ) ) : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> S  ->  ran  ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )  C_  S )
9088, 89syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  ran  ( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  ( f `
 z ) ) ) )  C_  S
)
91 cnrest2 17304 . . . . . . . . . 10  |-  ( ( J  e.  (TopOn `  CC )  /\  ran  (
z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( ( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  ( f `  z ) ) ) )  C_  S  /\  S  C_  CC )  -> 
( ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )  e.  ( ( II  tX  II )  Cn  J
)  <->  ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )  e.  ( ( II  tX  II )  Cn  ( Jt  S ) ) ) )
9227, 90, 34, 91syl3anc 1184 . . . . . . . . 9  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )  e.  ( ( II  tX  II )  Cn  J
)  <->  ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )  e.  ( ( II  tX  II )  Cn  ( Jt  S ) ) ) )
9365, 92mpbid 202 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  ( f `
 z ) ) ) )  e.  ( ( II  tX  II )  Cn  ( Jt  S ) ) )
944oveq2i 6051 . . . . . . . 8  |-  ( ( II  tX  II )  Cn  K )  =  ( ( II  tX  II )  Cn  ( Jt  S ) )
9593, 94syl6eleqr 2495 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  ( f `
 z ) ) ) )  e.  ( ( II  tX  II )  Cn  K ) )
96 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
97 simpr 448 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  0 )  ->  t  =  0 )
9897oveq1d 6055 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( t  x.  ( f `  0
) )  =  ( 0  x.  ( f `
 0 ) ) )
9997oveq2d 6056 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( 1  -  t )  =  ( 1  -  0 ) )
10047subid1i 9328 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
10199, 100syl6eq 2452 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( 1  -  t )  =  1 )
102 simpl 444 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  0 )  ->  z  =  s )
103102fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( f `  z )  =  ( f `  s ) )
104101, 103oveq12d 6058 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( 1  x.  ( f `
 s ) ) )
10598, 104oveq12d 6058 . . . . . . . . . 10  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( 0  x.  (
f `  0 )
)  +  ( 1  x.  ( f `  s ) ) ) )
106 ovex 6065 . . . . . . . . . 10  |-  ( ( 0  x.  ( f `
 0 ) )  +  ( 1  x.  ( f `  s
) ) )  e. 
_V
107105, 86, 106ovmpt2a 6163 . . . . . . . . 9  |-  ( ( s  e.  ( 0 [,] 1 )  /\  0  e.  ( 0 [,] 1 ) )  ->  ( s ( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( ( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  ( f `  z ) ) ) ) 0 )  =  ( ( 0  x.  ( f `  0
) )  +  ( 1  x.  ( f `
 s ) ) ) )
10896, 16, 107sylancl 644 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
s ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) 0 )  =  ( ( 0  x.  ( f `
 0 ) )  +  ( 1  x.  ( f `  s
) ) ) )
10942adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
f `  0 )  e.  CC )
110109mul02d 9220 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( f `
 0 ) )  =  0 )
11126toponunii 16952 . . . . . . . . . . . . 13  |-  CC  =  U. J
11213, 111cnf 17264 . . . . . . . . . . . 12  |-  ( f  e.  ( II  Cn  J )  ->  f : ( 0 [,] 1 ) --> CC )
11360, 112syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f : ( 0 [,] 1 ) --> CC )
114113ffvelrnda 5829 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
f `  s )  e.  CC )
115114mulid2d 9062 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( f `
 s ) )  =  ( f `  s ) )
116110, 115oveq12d 6058 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 0  x.  (
f `  0 )
)  +  ( 1  x.  ( f `  s ) ) )  =  ( 0  +  ( f `  s
) ) )
117114addid2d 9223 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  +  ( f `
 s ) )  =  ( f `  s ) )
118108, 116, 1173eqtrd 2440 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
s ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) 0 )  =  ( f `
 s ) )
119 1elunit 10972 . . . . . . . . 9  |-  1  e.  ( 0 [,] 1
)
120 simpr 448 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  t  =  1 )
121120oveq1d 6055 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( t  x.  ( f `  0
) )  =  ( 1  x.  ( f `
 0 ) ) )
122120oveq2d 6056 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( 1  -  t )  =  ( 1  -  1 ) )
123 1m1e0 10024 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
124122, 123syl6eq 2452 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( 1  -  t )  =  0 )
125 simpl 444 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  1 )  ->  z  =  s )
126125fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( f `  z )  =  ( f `  s ) )
127124, 126oveq12d 6058 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( 0  x.  ( f `
 s ) ) )
128121, 127oveq12d 6058 . . . . . . . . . 10  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( 1  x.  (
f `  0 )
)  +  ( 0  x.  ( f `  s ) ) ) )
129 ovex 6065 . . . . . . . . . 10  |-  ( ( 1  x.  ( f `
 0 ) )  +  ( 0  x.  ( f `  s
) ) )  e. 
_V
130128, 86, 129ovmpt2a 6163 . . . . . . . . 9  |-  ( ( s  e.  ( 0 [,] 1 )  /\  1  e.  ( 0 [,] 1 ) )  ->  ( s ( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( ( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  ( f `  z ) ) ) ) 1 )  =  ( ( 1  x.  ( f `  0
) )  +  ( 0  x.  ( f `
 s ) ) ) )
13196, 119, 130sylancl 644 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
s ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) 1 )  =  ( ( 1  x.  ( f `
 0 ) )  +  ( 0  x.  ( f `  s
) ) ) )
132109mulid2d 9062 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( f `
 0 ) )  =  ( f ` 
0 ) )
133114mul02d 9220 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( f `
 s ) )  =  0 )
134132, 133oveq12d 6058 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  x.  (
f `  0 )
)  +  ( 0  x.  ( f `  s ) ) )  =  ( ( f `
 0 )  +  0 ) )
135109addid1d 9222 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( f `  0
)  +  0 )  =  ( f ` 
0 ) )
136 fvex 5701 . . . . . . . . . . 11  |-  ( f `
 0 )  e. 
_V
137136fvconst2 5906 . . . . . . . . . 10  |-  ( s  e.  ( 0 [,] 1 )  ->  (
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) `  s )  =  ( f `  0 ) )
138137adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) `  s )  =  ( f `  0 ) )
139135, 138eqtr4d 2439 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( f `  0
)  +  0 )  =  ( ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) `  s ) )
140131, 134, 1393eqtrd 2440 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
s ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) 1 )  =  ( ( ( 0 [,] 1
)  X.  { ( f `  0 ) } ) `  s
) )
141 simpr 448 . . . . . . . . . . . 12  |-  ( ( z  =  0  /\  t  =  s )  ->  t  =  s )
142141oveq1d 6055 . . . . . . . . . . 11  |-  ( ( z  =  0  /\  t  =  s )  ->  ( t  x.  ( f `  0
) )  =  ( s  x.  ( f `
 0 ) ) )
143141oveq2d 6056 . . . . . . . . . . . 12  |-  ( ( z  =  0  /\  t  =  s )  ->  ( 1  -  t )  =  ( 1  -  s ) )
144 simpl 444 . . . . . . . . . . . . 13  |-  ( ( z  =  0  /\  t  =  s )  ->  z  =  0 )
145144fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( z  =  0  /\  t  =  s )  ->  ( f `  z )  =  ( f `  0 ) )
146143, 145oveq12d 6058 . . . . . . . . . . 11  |-  ( ( z  =  0  /\  t  =  s )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( ( 1  -  s
)  x.  ( f `
 0 ) ) )
147142, 146oveq12d 6058 . . . . . . . . . 10  |-  ( ( z  =  0  /\  t  =  s )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( s  x.  (
f `  0 )
)  +  ( ( 1  -  s )  x.  ( f ` 
0 ) ) ) )
148 ovex 6065 . . . . . . . . . 10  |-  ( ( s  x.  ( f `
 0 ) )  +  ( ( 1  -  s )  x.  ( f `  0
) ) )  e. 
_V
149147, 86, 148ovmpt2a 6163 . . . . . . . . 9  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 ( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( ( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  ( f `  z ) ) ) ) s )  =  ( ( s  x.  ( f `  0
) )  +  ( ( 1  -  s
)  x.  ( f `
 0 ) ) ) )
15016, 96, 149sylancr 645 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) s )  =  ( ( s  x.  ( f `
 0 ) )  +  ( ( 1  -  s )  x.  ( f `  0
) ) ) )
15130, 96sseldi 3306 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  CC )
152 pncan3 9269 . . . . . . . . . . 11  |-  ( ( s  e.  CC  /\  1  e.  CC )  ->  ( s  +  ( 1  -  s ) )  =  1 )
153151, 47, 152sylancl 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  +  ( 1  -  s ) )  =  1 )
154153oveq1d 6055 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s  +  ( 1  -  s ) )  x.  ( f `
 0 ) )  =  ( 1  x.  ( f `  0
) ) )
155 subcl 9261 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( 1  -  s
)  e.  CC )
15647, 151, 155sylancr 645 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  -  s )  e.  CC )
157151, 156, 109adddird 9069 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s  +  ( 1  -  s ) )  x.  ( f `
 0 ) )  =  ( ( s  x.  ( f ` 
0 ) )  +  ( ( 1  -  s )  x.  (
f `  0 )
) ) )
158154, 157, 1323eqtr3d 2444 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s  x.  (
f `  0 )
)  +  ( ( 1  -  s )  x.  ( f ` 
0 ) ) )  =  ( f ` 
0 ) )
159150, 158eqtrd 2436 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) s )  =  ( f `
 0 ) )
160 simpr 448 . . . . . . . . . . . 12  |-  ( ( z  =  1  /\  t  =  s )  ->  t  =  s )
161160oveq1d 6055 . . . . . . . . . . 11  |-  ( ( z  =  1  /\  t  =  s )  ->  ( t  x.  ( f `  0
) )  =  ( s  x.  ( f `
 0 ) ) )
162160oveq2d 6056 . . . . . . . . . . . 12  |-  ( ( z  =  1  /\  t  =  s )  ->  ( 1  -  t )  =  ( 1  -  s ) )
163 simpl 444 . . . . . . . . . . . . 13  |-  ( ( z  =  1  /\  t  =  s )  ->  z  =  1 )
164163fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( z  =  1  /\  t  =  s )  ->  ( f `  z )  =  ( f `  1 ) )
165162, 164oveq12d 6058 . . . . . . . . . . 11  |-  ( ( z  =  1  /\  t  =  s )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( ( 1  -  s
)  x.  ( f `
 1 ) ) )
166161, 165oveq12d 6058 . . . . . . . . . 10  |-  ( ( z  =  1  /\  t  =  s )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( s  x.  (
f `  0 )
)  +  ( ( 1  -  s )  x.  ( f ` 
1 ) ) ) )
167 ovex 6065 . . . . . . . . . 10  |-  ( ( s  x.  ( f `
 0 ) )  +  ( ( 1  -  s )  x.  ( f `  1
) ) )  e. 
_V
168166, 86, 167ovmpt2a 6163 . . . . . . . . 9  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 ( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( ( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  ( f `  z ) ) ) ) s )  =  ( ( s  x.  ( f `  0
) )  +  ( ( 1  -  s
)  x.  ( f `
 1 ) ) ) )
169119, 96, 168sylancr 645 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) s )  =  ( ( s  x.  ( f `
 0 ) )  +  ( ( 1  -  s )  x.  ( f `  1
) ) ) )
170 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
f `  0 )  =  ( f ` 
1 ) )
171170oveq2d 6056 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( 1  -  s
)  x.  ( f `
 0 ) )  =  ( ( 1  -  s )  x.  ( f `  1
) ) )
172171oveq2d 6056 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s  x.  (
f `  0 )
)  +  ( ( 1  -  s )  x.  ( f ` 
0 ) ) )  =  ( ( s  x.  ( f ` 
0 ) )  +  ( ( 1  -  s )  x.  (
f `  1 )
) ) )
173158, 172, 1703eqtr3d 2444 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( s  x.  (
f `  0 )
)  +  ( ( 1  -  s )  x.  ( f ` 
1 ) ) )  =  ( f ` 
1 ) )
174169, 173eqtrd 2436 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 ( z  e.  ( 0 [,] 1
) ,  t  e.  ( 0 [,] 1
)  |->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) ) s )  =  ( f `
 1 ) )
1756, 22, 95, 118, 140, 159, 174isphtpy2d 18965 . . . . . 6  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 ) 
|->  ( ( t  x.  ( f `  0
) )  +  ( ( 1  -  t
)  x.  ( f `
 z ) ) ) )  e.  ( f ( PHtpy `  K
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) )
176 ne0i 3594 . . . . . 6  |-  ( ( z  e.  ( 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( ( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  ( f `  z ) ) ) )  e.  ( f ( PHtpy `  K )
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) )  ->  ( f (
PHtpy `  K ) ( ( 0 [,] 1
)  X.  { ( f `  0 ) } ) )  =/=  (/) )
177175, 176syl 16 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f ( PHtpy `  K ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) )  =/=  (/) )
178 isphtpc 18972 . . . . 5  |-  ( f (  ~=ph  `  K ) ( ( 0 [,] 1 )  X.  {
( f `  0
) } )  <->  ( f  e.  ( II  Cn  K
)  /\  ( (
0 [,] 1 )  X.  { ( f `
 0 ) } )  e.  ( II 
Cn  K )  /\  ( f ( PHtpy `  K ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) )  =/=  (/) ) )
1796, 22, 177, 178syl3anbrc 1138 . . . 4  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f (  ~=ph  `  K
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) )
180179expr 599 . . 3  |-  ( (
ph  /\  f  e.  ( II  Cn  K
) )  ->  (
( f `  0
)  =  ( f `
 1 )  -> 
f (  ~=ph  `  K
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) )
181180ralrimiva 2749 . 2  |-  ( ph  ->  A. f  e.  ( II  Cn  K ) ( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  K ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ) )
182 isscon 24866 . 2  |-  ( K  e. SCon 
<->  ( K  e. PCon  /\  A. f  e.  ( II 
Cn  K ) ( ( f `  0
)  =  ( f `
 1 )  -> 
f (  ~=ph  `  K
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) ) )
1835, 181, 182sylanbrc 646 1  |-  ( ph  ->  K  e. SCon )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666    C_ wss 3280   (/)c0 3588   {csn 3774   U.cuni 3975   class class class wbr 4172    X. cxp 4835   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   [,]cicc 10875   ↾t crest 13603   TopOpenctopn 13604  ℂfldccnfld 16658   Topctop 16913  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545   IIcii 18858   PHtpycphtpy 18946    ~=ph cphtpc 18947  PConcpcon 24859  SConcscon 24860
This theorem is referenced by:  blscon  24884  rescon  24886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860  df-htpy 18948  df-phtpy 18949  df-phtpc 18970  df-pcon 24861  df-scon 24862
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