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Theorem cvxscon 24922
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 24921 . 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 24904 . . . . . . . . . 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 2435 . . . . . . . . 9  |-  U. K  =  U. K
1110toptopon 16990 . . . . . . . 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 18903 . . . . . . . . . 10  |-  ( 0 [,] 1 )  = 
U. II
1413, 10cnf 17302 . . . . . . . . 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 11007 . . . . . . . 8  |-  0  e.  ( 0 [,] 1
)
17 ffvelrn 5860 . . . . . . . 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 2435 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )
2019pcoptcl 19038 . . . . . . 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 18901 . . . . . . . . . . 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 18905 . . . . . . . . . . . 12  |-  II  =  ( Jt  ( 0 [,] 1 ) )
263cnfldtopon 18809 . . . . . . . . . . . . 13  |-  J  e.  (TopOn `  CC )
2726a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  J  e.  (TopOn `  CC ) )
28 unitssre 11034 . . . . . . . . . . . . . 14  |-  ( 0 [,] 1 )  C_  RR
29 ax-resscn 9039 . . . . . . . . . . . . . 14  |-  RR  C_  CC
3028, 29sstri 3349 . . . . . . . . . . . . 13  |-  ( 0 [,] 1 )  C_  CC
3130a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( 0 [,] 1
)  C_  CC )
3227, 27cnmpt2nd 17693 . . . . . . . . . . . 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 17701 . . . . . . . . . . 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 17217 . . . . . . . . . . . . . . . . . 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 2519 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  K  e.  (TopOn `  S ) )
38 toponuni 16984 . . . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f `  0
)  e.  S )
4234, 41sseldd 3341 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f `  0
)  e.  CC )
4324, 24, 27, 42cnmpt2c 17694 . . . . . . . . . . 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 18889 . . . . . . . . . . . 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 17699 . . . . . . . . . 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 9040 . . . . . . . . . . . . . . 15  |-  1  e.  CC
4847a1i 11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
1  e.  CC )
4927, 27, 27, 48cnmpt2c 17694 . . . . . . . . . . . . 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 18888 . . . . . . . . . . . . . 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 17699 . . . . . . . . . . . 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 17701 . . . . . . . . . . 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 17692 . . . . . . . . . . . 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 18810 . . . . . . . . . . . . . 14  |-  J  e. 
Top
56 cnrest2r 17343 . . . . . . . . . . . . . 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 6084 . . . . . . . . . . . . . 14  |-  ( II 
Cn  K )  =  ( II  Cn  ( Jt  S ) )
596, 58syl6eleq 2525 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f  e.  ( II 
Cn  ( Jt  S ) ) )
6057, 59sseldi 3338 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f  e.  ( II 
Cn  J ) )
6124, 24, 54, 60cnmpt21f 17696 . . . . . . . . . . 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 17699 . . . . . . . . . 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 18887 . . . . . . . . . . 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 17699 . . . . . . . . 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 5863 . . . . . . . . . . . . . . 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 2512 . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . . . 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 6081 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( f ` 
0 )  ->  (
t  x.  x )  =  ( t  x.  ( f `  0
) ) )
7877oveq1d 6088 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( f ` 
0 )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  =  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  y
) ) )
7978eleq1d 2501 . . . . . . . . . . . . . . 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 6081 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( f `  z )  ->  (
( 1  -  t
)  x.  y )  =  ( ( 1  -  t )  x.  ( f `  z
) ) )
8180oveq2d 6089 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( f `  z )  ->  (
( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  y ) )  =  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )
8281eleq1d 2501 . . . . . . . . . . . . . . 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 3051 . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . 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 2435 . . . . . . . . . . . . 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 6410 . . . . . . . . . . . 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 5589 . . . . . . . . . . 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 17342 . . . . . . . . . 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 6084 . . . . . . . 8  |-  ( ( II  tX  II )  Cn  K )  =  ( ( II  tX  II )  Cn  ( Jt  S ) )
9593, 94syl6eleqr 2526 . . . . . . 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 6088 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( t  x.  ( f `  0
) )  =  ( 0  x.  ( f `
 0 ) ) )
9997oveq2d 6089 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( 1  -  t )  =  ( 1  -  0 ) )
10047subid1i 9364 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
10199, 100syl6eq 2483 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( 1  -  t )  =  1 )
102 simpl 444 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  0 )  ->  z  =  s )
103102fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( f `  z )  =  ( f `  s ) )
104101, 103oveq12d 6091 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( 1  x.  ( f `
 s ) ) )
10598, 104oveq12d 6091 . . . . . . . . . 10  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( 0  x.  (
f `  0 )
)  +  ( 1  x.  ( f `  s ) ) ) )
106 ovex 6098 . . . . . . . . . 10  |-  ( ( 0  x.  ( f `
 0 ) )  +  ( 1  x.  ( f `  s
) ) )  e. 
_V
107105, 86, 106ovmpt2a 6196 . . . . . . . . 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 9256 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( f `
 0 ) )  =  0 )
11126toponunii 16989 . . . . . . . . . . . . 13  |-  CC  =  U. J
11213, 111cnf 17302 . . . . . . . . . . . 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 5862 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
f `  s )  e.  CC )
115114mulid2d 9098 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( f `
 s ) )  =  ( f `  s ) )
116110, 115oveq12d 6091 . . . . . . . 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 9259 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  +  ( f `
 s ) )  =  ( f `  s ) )
118108, 116, 1173eqtrd 2471 . . . . . . 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 11008 . . . . . . . . 9  |-  1  e.  ( 0 [,] 1
)
120 simpr 448 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  t  =  1 )
121120oveq1d 6088 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( t  x.  ( f `  0
) )  =  ( 1  x.  ( f `
 0 ) ) )
122120oveq2d 6089 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( 1  -  t )  =  ( 1  -  1 ) )
123 1m1e0 10060 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
124122, 123syl6eq 2483 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( 1  -  t )  =  0 )
125 simpl 444 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  1 )  ->  z  =  s )
126125fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( f `  z )  =  ( f `  s ) )
127124, 126oveq12d 6091 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( 0  x.  ( f `
 s ) ) )
128121, 127oveq12d 6091 . . . . . . . . . 10  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( 1  x.  (
f `  0 )
)  +  ( 0  x.  ( f `  s ) ) ) )
129 ovex 6098 . . . . . . . . . 10  |-  ( ( 1  x.  ( f `
 0 ) )  +  ( 0  x.  ( f `  s
) ) )  e. 
_V
130128, 86, 129ovmpt2a 6196 . . . . . . . . 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 9098 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( f `
 0 ) )  =  ( f ` 
0 ) )
133114mul02d 9256 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( f `
 s ) )  =  0 )
134132, 133oveq12d 6091 . . . . . . . 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 9258 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( f `  0
)  +  0 )  =  ( f ` 
0 ) )
136 fvex 5734 . . . . . . . . . . 11  |-  ( f `
 0 )  e. 
_V
137136fvconst2 5939 . . . . . . . . . 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 2470 . . . . . . . 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 2471 . . . . . . 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 6088 . . . . . . . . . . 11  |-  ( ( z  =  0  /\  t  =  s )  ->  ( t  x.  ( f `  0
) )  =  ( s  x.  ( f `
 0 ) ) )
143141oveq2d 6089 . . . . . . . . . . . 12  |-  ( ( z  =  0  /\  t  =  s )  ->  ( 1  -  t )  =  ( 1  -  s ) )
144 simpl 444 . . . . . . . . . . . . 13  |-  ( ( z  =  0  /\  t  =  s )  ->  z  =  0 )
145144fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( z  =  0  /\  t  =  s )  ->  ( f `  z )  =  ( f `  0 ) )
146143, 145oveq12d 6091 . . . . . . . . . . 11  |-  ( ( z  =  0  /\  t  =  s )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( ( 1  -  s
)  x.  ( f `
 0 ) ) )
147142, 146oveq12d 6091 . . . . . . . . . 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 6098 . . . . . . . . . 10  |-  ( ( s  x.  ( f `
 0 ) )  +  ( ( 1  -  s )  x.  ( f `  0
) ) )  e. 
_V
149147, 86, 148ovmpt2a 6196 . . . . . . . . 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 3338 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  CC )
152 pncan3 9305 . . . . . . . . . . 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 6088 . . . . . . . . 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 9297 . . . . . . . . . . 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 9105 . . . . . . . . 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 2475 . . . . . . . 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 2467 . . . . . . 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 6088 . . . . . . . . . . 11  |-  ( ( z  =  1  /\  t  =  s )  ->  ( t  x.  ( f `  0
) )  =  ( s  x.  ( f `
 0 ) ) )
162160oveq2d 6089 . . . . . . . . . . . 12  |-  ( ( z  =  1  /\  t  =  s )  ->  ( 1  -  t )  =  ( 1  -  s ) )
163 simpl 444 . . . . . . . . . . . . 13  |-  ( ( z  =  1  /\  t  =  s )  ->  z  =  1 )
164163fveq2d 5724 . . . . . . . . . . . 12  |-  ( ( z  =  1  /\  t  =  s )  ->  ( f `  z )  =  ( f `  1 ) )
165162, 164oveq12d 6091 . . . . . . . . . . 11  |-  ( ( z  =  1  /\  t  =  s )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( ( 1  -  s
)  x.  ( f `
 1 ) ) )
166161, 165oveq12d 6091 . . . . . . . . . 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 6098 . . . . . . . . . 10  |-  ( ( s  x.  ( f `
 0 ) )  +  ( ( 1  -  s )  x.  ( f `  1
) ) )  e. 
_V
168166, 86, 167ovmpt2a 6196 . . . . . . . . 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 6089 . . . . . . . . . 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 6089 . . . . . . . . 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 2475 . . . . . . . 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 2467 . . . . . . 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 19004 . . . . . 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 3626 . . . . . 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 19011 . . . . 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 2781 . 2  |-  ( ph  ->  A. f  e.  ( II  Cn  K ) ( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  K ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ) )
182 isscon 24905 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    C_ wss 3312   (/)c0 3620   {csn 3806   U.cuni 4007   class class class wbr 4204    X. cxp 4868   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   [,]cicc 10911   ↾t crest 13640   TopOpenctopn 13641  ℂfldccnfld 16695   Topctop 16950  TopOnctopon 16951    Cn ccn 17280    tX ctx 17584   IIcii 18897   PHtpycphtpy 18985    ~=ph cphtpc 18986  PConcpcon 24898  SConcscon 24899
This theorem is referenced by:  blscon  24923  rescon  24925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cn 17283  df-cnp 17284  df-tx 17586  df-hmeo 17779  df-xms 18342  df-ms 18343  df-tms 18344  df-ii 18899  df-htpy 18987  df-phtpy 18988  df-phtpc 19009  df-pcon 24900  df-scon 24901
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