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Theorem cvxscon 23789
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 23788 . 2  |-  ( ph  ->  K  e. PCon )
6 simprl 732 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f  e.  ( II 
Cn  K ) )
7 pcontop 23771 . . . . . . . . . 10  |-  ( K  e. PCon  ->  K  e.  Top )
85, 7syl 15 . . . . . . . . 9  |-  ( ph  ->  K  e.  Top )
98adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  K  e.  Top )
10 eqid 2296 . . . . . . . . 9  |-  U. K  =  U. K
1110toptopon 16687 . . . . . . . 8  |-  ( K  e.  Top  <->  K  e.  (TopOn `  U. K ) )
129, 11sylib 188 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  K  e.  (TopOn `  U. K ) )
13 iiuni 18401 . . . . . . . . . 10  |-  ( 0 [,] 1 )  = 
U. II
1413, 10cnf 16992 . . . . . . . . 9  |-  ( f  e.  ( II  Cn  K )  ->  f : ( 0 [,] 1 ) --> U. K
)
156, 14syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f : ( 0 [,] 1 ) --> U. K )
16 0elunit 10770 . . . . . . . 8  |-  0  e.  ( 0 [,] 1
)
17 ffvelrn 5679 . . . . . . . 8  |-  ( ( f : ( 0 [,] 1 ) --> U. K  /\  0  e.  ( 0 [,] 1
) )  ->  (
f `  0 )  e.  U. K )
1815, 16, 17sylancl 643 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f `  0
)  e.  U. K
)
19 eqid 2296 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } )
2019pcoptcl 18535 . . . . . . 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 642 . . . . . 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 967 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( ( 0 [,] 1 )  X.  {
( f `  0
) } )  e.  ( II  Cn  K
) )
23 iitopon 18399 . . . . . . . . . . 11  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
2423a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  II  e.  (TopOn `  (
0 [,] 1 ) ) )
253dfii3 18403 . . . . . . . . . . . 12  |-  II  =  ( Jt  ( 0 [,] 1 ) )
263cnfldtopon 18308 . . . . . . . . . . . . 13  |-  J  e.  (TopOn `  CC )
2726a1i 10 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  J  e.  (TopOn `  CC ) )
28 0re 8854 . . . . . . . . . . . . . . 15  |-  0  e.  RR
29 1re 8853 . . . . . . . . . . . . . . 15  |-  1  e.  RR
30 iccssre 10747 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
3128, 29, 30mp2an 653 . . . . . . . . . . . . . 14  |-  ( 0 [,] 1 )  C_  RR
32 ax-resscn 8810 . . . . . . . . . . . . . 14  |-  RR  C_  CC
3331, 32sstri 3201 . . . . . . . . . . . . 13  |-  ( 0 [,] 1 )  C_  CC
3433a1i 10 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( 0 [,] 1
)  C_  CC )
3527, 27cnmpt2nd 17379 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  CC ,  t  e.  CC  |->  t )  e.  ( ( J  tX  J
)  Cn  J ) )
3625, 27, 34, 25, 27, 34, 35cnmpt2res 17387 . . . . . . . . . . 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 ) )
371adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  S  C_  CC )
38 resttopon 16908 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  CC )  /\  S  C_  CC )  ->  ( Jt  S )  e.  (TopOn `  S ) )
3926, 1, 38sylancr 644 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( Jt  S )  e.  (TopOn `  S ) )
404, 39syl5eqel 2380 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  K  e.  (TopOn `  S ) )
41 toponuni 16681 . . . . . . . . . . . . . . . 16  |-  ( K  e.  (TopOn `  S
)  ->  S  =  U. K )
4240, 41syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  S  =  U. K
)
4342adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  S  =  U. K )
4418, 43eleqtrrd 2373 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f `  0
)  e.  S )
4537, 44sseldd 3194 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f `  0
)  e.  CC )
4624, 24, 27, 45cnmpt2c 17380 . . . . . . . . . . 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 ) )
473mulcn 18387 . . . . . . . . . . . 12  |-  x.  e.  ( ( J  tX  J )  Cn  J
)
4847a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  x.  e.  ( ( J 
tX  J )  Cn  J ) )
4924, 24, 36, 46, 48cnmpt22f 17385 . . . . . . . . . 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 ) )
50 ax-1cn 8811 . . . . . . . . . . . . . . 15  |-  1  e.  CC
5150a1i 10 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
1  e.  CC )
5227, 27, 27, 51cnmpt2c 17380 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( z  e.  CC ,  t  e.  CC  |->  1 )  e.  ( ( J  tX  J
)  Cn  J ) )
533subcn 18386 . . . . . . . . . . . . . 14  |-  -  e.  ( ( J  tX  J )  Cn  J
)
5453a1i 10 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  -  e.  ( ( J  tX  J )  Cn  J ) )
5527, 27, 52, 35, 54cnmpt22f 17385 . . . . . . . . . . . 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 ) )
5625, 27, 34, 25, 27, 34, 55cnmpt2res 17387 . . . . . . . . . . 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 ) )
5724, 24cnmpt1st 17378 . . . . . . . . . . . 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 ) )
583cnfldtop 18309 . . . . . . . . . . . . . 14  |-  J  e. 
Top
59 cnrest2r 17031 . . . . . . . . . . . . . 14  |-  ( J  e.  Top  ->  (
II  Cn  ( Jt  S
) )  C_  (
II  Cn  J )
)
6058, 59ax-mp 8 . . . . . . . . . . . . 13  |-  ( II 
Cn  ( Jt  S ) )  C_  ( II  Cn  J )
614oveq2i 5885 . . . . . . . . . . . . . 14  |-  ( II 
Cn  K )  =  ( II  Cn  ( Jt  S ) )
626, 61syl6eleq 2386 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f  e.  ( II 
Cn  ( Jt  S ) ) )
6360, 62sseldi 3191 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f  e.  ( II 
Cn  J ) )
6424, 24, 57, 63cnmpt21f 17382 . . . . . . . . . . 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 ) )
6524, 24, 56, 64, 48cnmpt22f 17385 . . . . . . . . . 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 ) )
663addcn 18385 . . . . . . . . . . 11  |-  +  e.  ( ( J  tX  J )  Cn  J
)
6766a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  ->  +  e.  ( ( J  tX  J )  Cn  J ) )
6824, 24, 49, 65, 67cnmpt22f 17385 . . . . . . . . 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 ) )
6944adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( f `  0
)  e.  S )
7015adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
f : ( 0 [,] 1 ) --> U. K )
71 simprl 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
z  e.  ( 0 [,] 1 ) )
72 ffvelrn 5679 . . . . . . . . . . . . . . . 16  |-  ( ( f : ( 0 [,] 1 ) --> U. K  /\  z  e.  ( 0 [,] 1
) )  ->  (
f `  z )  e.  U. K )
7370, 71, 72syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( f `  z
)  e.  U. K
)
7443adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  ->  S  =  U. K )
7573, 74eleqtrrd 2373 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  ( z  e.  ( 0 [,] 1 )  /\  t  e.  ( 0 [,] 1
) ) )  -> 
( f `  z
)  e.  S )
7623exp2 1169 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( x  e.  S  ->  ( y  e.  S  ->  ( t  e.  ( 0 [,] 1 )  ->  ( ( t  x.  x )  +  ( ( 1  -  t )  x.  y
) )  e.  S
) ) ) )
7776imp42 577 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  S  /\  y  e.  S )
)  /\  t  e.  ( 0 [,] 1
) )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  e.  S )
7877an32s 779 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  t  e.  ( 0 [,] 1
) )  /\  (
x  e.  S  /\  y  e.  S )
)  ->  ( (
t  x.  x )  +  ( ( 1  -  t )  x.  y ) )  e.  S )
7978ralrimivva 2648 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  t  e.  ( 0 [,] 1
) )  ->  A. x  e.  S  A. y  e.  S  ( (
t  x.  x )  +  ( ( 1  -  t )  x.  y ) )  e.  S )
8079ad2ant2rl 729 . . . . . . . . . . . . . 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 )
81 oveq2 5882 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( f ` 
0 )  ->  (
t  x.  x )  =  ( t  x.  ( f `  0
) ) )
8281oveq1d 5889 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( f ` 
0 )  ->  (
( t  x.  x
)  +  ( ( 1  -  t )  x.  y ) )  =  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  y
) ) )
8382eleq1d 2362 . . . . . . . . . . . . . . 15  |-  ( x  =  ( f ` 
0 )  ->  (
( ( t  x.  x )  +  ( ( 1  -  t
)  x.  y ) )  e.  S  <->  ( (
t  x.  ( f `
 0 ) )  +  ( ( 1  -  t )  x.  y ) )  e.  S ) )
84 oveq2 5882 . . . . . . . . . . . . . . . . 17  |-  ( y  =  ( f `  z )  ->  (
( 1  -  t
)  x.  y )  =  ( ( 1  -  t )  x.  ( f `  z
) ) )
8584oveq2d 5890 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( f `  z )  ->  (
( t  x.  (
f `  0 )
)  +  ( ( 1  -  t )  x.  y ) )  =  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) ) )
8685eleq1d 2362 . . . . . . . . . . . . . . 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 ) )
8783, 86rspc2va 2904 . . . . . . . . . . . . . 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 )
8869, 75, 80, 87syl21anc 1181 . . . . . . . . . . . . 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 )
8988ralrimivva 2648 . . . . . . . . . . . 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 )
90 eqid 2296 . . . . . . . . . . . . 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 )
) ) )
9190fmpt2 6207 . . . . . . . . . . . 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 )
9289, 91sylib 188 . . . . . . . . . . 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 )
93 frn 5411 . . . . . . . . . . 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 )
9492, 93syl 15 . . . . . . . . . 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
)
95 cnrest2 17030 . . . . . . . . . 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 ) ) ) )
9627, 94, 37, 95syl3anc 1182 . . . . . . . . 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 ) ) ) )
9768, 96mpbid 201 . . . . . . . 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 ) ) )
984oveq2i 5885 . . . . . . . 8  |-  ( ( II  tX  II )  Cn  K )  =  ( ( II  tX  II )  Cn  ( Jt  S ) )
9997, 98syl6eleqr 2387 . . . . . . 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 ) )
100 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
101 simpr 447 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  0 )  ->  t  =  0 )
102101oveq1d 5889 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( t  x.  ( f `  0
) )  =  ( 0  x.  ( f `
 0 ) ) )
103101oveq2d 5890 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( 1  -  t )  =  ( 1  -  0 ) )
10450subid1i 9134 . . . . . . . . . . . . 13  |-  ( 1  -  0 )  =  1
105103, 104syl6eq 2344 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( 1  -  t )  =  1 )
106 simpl 443 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  0 )  ->  z  =  s )
107106fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( f `  z )  =  ( f `  s ) )
108105, 107oveq12d 5892 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( 1  x.  ( f `
 s ) ) )
109102, 108oveq12d 5892 . . . . . . . . . 10  |-  ( ( z  =  s  /\  t  =  0 )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( 0  x.  (
f `  0 )
)  +  ( 1  x.  ( f `  s ) ) ) )
110 ovex 5899 . . . . . . . . . 10  |-  ( ( 0  x.  ( f `
 0 ) )  +  ( 1  x.  ( f `  s
) ) )  e. 
_V
111109, 90, 110ovmpt2a 5994 . . . . . . . . 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 ) ) ) )
112100, 16, 111sylancl 643 . . . . . . . 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
) ) ) )
11345adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
f `  0 )  e.  CC )
114113mul02d 9026 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( f `
 0 ) )  =  0 )
11526toponunii 16686 . . . . . . . . . . . . 13  |-  CC  =  U. J
11613, 115cnf 16992 . . . . . . . . . . . 12  |-  ( f  e.  ( II  Cn  J )  ->  f : ( 0 [,] 1 ) --> CC )
11763, 116syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f : ( 0 [,] 1 ) --> CC )
118 ffvelrn 5679 . . . . . . . . . . 11  |-  ( ( f : ( 0 [,] 1 ) --> CC 
/\  s  e.  ( 0 [,] 1 ) )  ->  ( f `  s )  e.  CC )
119117, 118sylan 457 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
f `  s )  e.  CC )
120119mulid2d 8869 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( f `
 s ) )  =  ( f `  s ) )
121114, 120oveq12d 5892 . . . . . . . 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
) ) )
122119addid2d 9029 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  +  ( f `
 s ) )  =  ( f `  s ) )
123112, 121, 1223eqtrd 2332 . . . . . . 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 ) )
124 1elunit 10771 . . . . . . . . 9  |-  1  e.  ( 0 [,] 1
)
125 simpr 447 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  t  =  1 )
126125oveq1d 5889 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( t  x.  ( f `  0
) )  =  ( 1  x.  ( f `
 0 ) ) )
127125oveq2d 5890 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( 1  -  t )  =  ( 1  -  1 ) )
128 1m1e0 9830 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
129127, 128syl6eq 2344 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( 1  -  t )  =  0 )
130 simpl 443 . . . . . . . . . . . . 13  |-  ( ( z  =  s  /\  t  =  1 )  ->  z  =  s )
131130fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( f `  z )  =  ( f `  s ) )
132129, 131oveq12d 5892 . . . . . . . . . . 11  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( 0  x.  ( f `
 s ) ) )
133126, 132oveq12d 5892 . . . . . . . . . 10  |-  ( ( z  =  s  /\  t  =  1 )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( 1  x.  (
f `  0 )
)  +  ( 0  x.  ( f `  s ) ) ) )
134 ovex 5899 . . . . . . . . . 10  |-  ( ( 1  x.  ( f `
 0 ) )  +  ( 0  x.  ( f `  s
) ) )  e. 
_V
135133, 90, 134ovmpt2a 5994 . . . . . . . . 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 ) ) ) )
136100, 124, 135sylancl 643 . . . . . . . 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
) ) ) )
137113mulid2d 8869 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  x.  ( f `
 0 ) )  =  ( f ` 
0 ) )
138119mul02d 9026 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
0  x.  ( f `
 s ) )  =  0 )
139137, 138oveq12d 5892 . . . . . . . 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 ) )
140113addid1d 9028 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( f `  0
)  +  0 )  =  ( f ` 
0 ) )
141 fvex 5555 . . . . . . . . . . 11  |-  ( f `
 0 )  e. 
_V
142141fvconst2 5745 . . . . . . . . . 10  |-  ( s  e.  ( 0 [,] 1 )  ->  (
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) `  s )  =  ( f `  0 ) )
143142adantl 452 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( ( 0 [,] 1 )  X.  {
( f `  0
) } ) `  s )  =  ( f `  0 ) )
144140, 143eqtr4d 2331 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
( f `  0
)  +  0 )  =  ( ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) `  s ) )
145136, 139, 1443eqtrd 2332 . . . . . . 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
) )
146 simpr 447 . . . . . . . . . . . 12  |-  ( ( z  =  0  /\  t  =  s )  ->  t  =  s )
147146oveq1d 5889 . . . . . . . . . . 11  |-  ( ( z  =  0  /\  t  =  s )  ->  ( t  x.  ( f `  0
) )  =  ( s  x.  ( f `
 0 ) ) )
148146oveq2d 5890 . . . . . . . . . . . 12  |-  ( ( z  =  0  /\  t  =  s )  ->  ( 1  -  t )  =  ( 1  -  s ) )
149 simpl 443 . . . . . . . . . . . . 13  |-  ( ( z  =  0  /\  t  =  s )  ->  z  =  0 )
150149fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( z  =  0  /\  t  =  s )  ->  ( f `  z )  =  ( f `  0 ) )
151148, 150oveq12d 5892 . . . . . . . . . . 11  |-  ( ( z  =  0  /\  t  =  s )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( ( 1  -  s
)  x.  ( f `
 0 ) ) )
152147, 151oveq12d 5892 . . . . . . . . . 10  |-  ( ( z  =  0  /\  t  =  s )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( s  x.  (
f `  0 )
)  +  ( ( 1  -  s )  x.  ( f ` 
0 ) ) ) )
153 ovex 5899 . . . . . . . . . 10  |-  ( ( s  x.  ( f `
 0 ) )  +  ( ( 1  -  s )  x.  ( f `  0
) ) )  e. 
_V
154152, 90, 153ovmpt2a 5994 . . . . . . . . 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 ) ) ) )
15516, 100, 154sylancr 644 . . . . . . . 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
) ) ) )
15633, 100sseldi 3191 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  CC )
157 pncan3 9075 . . . . . . . . . . 11  |-  ( ( s  e.  CC  /\  1  e.  CC )  ->  ( s  +  ( 1  -  s ) )  =  1 )
158156, 50, 157sylancl 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
s  +  ( 1  -  s ) )  =  1 )
159158oveq1d 5889 . . . . . . . . 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
) ) )
160 subcl 9067 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  s  e.  CC )  ->  ( 1  -  s
)  e.  CC )
16150, 156, 160sylancr 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
1  -  s )  e.  CC )
162156, 161, 113adddird 8876 . . . . . . . . 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 )
) ) )
163159, 162, 1373eqtr3d 2336 . . . . . . . 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 ) )
164155, 163eqtrd 2328 . . . . . . 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 ) )
165 simpr 447 . . . . . . . . . . . 12  |-  ( ( z  =  1  /\  t  =  s )  ->  t  =  s )
166165oveq1d 5889 . . . . . . . . . . 11  |-  ( ( z  =  1  /\  t  =  s )  ->  ( t  x.  ( f `  0
) )  =  ( s  x.  ( f `
 0 ) ) )
167165oveq2d 5890 . . . . . . . . . . . 12  |-  ( ( z  =  1  /\  t  =  s )  ->  ( 1  -  t )  =  ( 1  -  s ) )
168 simpl 443 . . . . . . . . . . . . 13  |-  ( ( z  =  1  /\  t  =  s )  ->  z  =  1 )
169168fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( z  =  1  /\  t  =  s )  ->  ( f `  z )  =  ( f `  1 ) )
170167, 169oveq12d 5892 . . . . . . . . . . 11  |-  ( ( z  =  1  /\  t  =  s )  ->  ( ( 1  -  t )  x.  ( f `  z
) )  =  ( ( 1  -  s
)  x.  ( f `
 1 ) ) )
171166, 170oveq12d 5892 . . . . . . . . . 10  |-  ( ( z  =  1  /\  t  =  s )  ->  ( ( t  x.  ( f ` 
0 ) )  +  ( ( 1  -  t )  x.  (
f `  z )
) )  =  ( ( s  x.  (
f `  0 )
)  +  ( ( 1  -  s )  x.  ( f ` 
1 ) ) ) )
172 ovex 5899 . . . . . . . . . 10  |-  ( ( s  x.  ( f `
 0 ) )  +  ( ( 1  -  s )  x.  ( f `  1
) ) )  e. 
_V
173171, 90, 172ovmpt2a 5994 . . . . . . . . 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 ) ) ) )
174124, 100, 173sylancr 644 . . . . . . . 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
) ) ) )
175 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  ( II 
Cn  K )  /\  ( f `  0
)  =  ( f `
 1 ) ) )  /\  s  e.  ( 0 [,] 1
) )  ->  (
f `  0 )  =  ( f ` 
1 ) )
176175oveq2d 5890 . . . . . . . . . 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
) ) )
177176oveq2d 5890 . . . . . . . . 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 )
) ) )
178163, 177, 1753eqtr3d 2336 . . . . . . . 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 ) )
179174, 178eqtrd 2328 . . . . . . 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 ) )
1806, 22, 99, 123, 145, 164, 179isphtpy2d 18501 . . . . . 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 ) } ) ) )
181 ne0i 3474 . . . . . 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 ) } ) )  =/=  (/) )
182180, 181syl 15 . . . . 5  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
( f ( PHtpy `  K ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) )  =/=  (/) )
183 isphtpc 18508 . . . . 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 ) } ) )  =/=  (/) ) )
1846, 22, 182, 183syl3anbrc 1136 . . . 4  |-  ( (
ph  /\  ( f  e.  ( II  Cn  K
)  /\  ( f `  0 )  =  ( f `  1
) ) )  -> 
f (  ~=ph  `  K
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) )
185184expr 598 . . 3  |-  ( (
ph  /\  f  e.  ( II  Cn  K
) )  ->  (
( f `  0
)  =  ( f `
 1 )  -> 
f (  ~=ph  `  K
) ( ( 0 [,] 1 )  X. 
{ ( f ` 
0 ) } ) ) )
186185ralrimiva 2639 . 2  |-  ( ph  ->  A. f  e.  ( II  Cn  K ) ( ( f ` 
0 )  =  ( f `  1 )  ->  f (  ~=ph  `  K ) ( ( 0 [,] 1 )  X.  { ( f `
 0 ) } ) ) )
187 isscon 23772 . 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 ) } ) ) ) )
1885, 186, 187sylanbrc 645 1  |-  ( ph  ->  K  e. SCon )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   class class class wbr 4039    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   [,]cicc 10675   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393   Topctop 16647  TopOnctopon 16648    Cn ccn 16970    tX ctx 17271   IIcii 18395   PHtpycphtpy 18482    ~=ph cphtpc 18483  PConcpcon 23765  SConcscon 23766
This theorem is referenced by:  blscon  23790  rescon  23792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-ii 18397  df-htpy 18484  df-phtpy 18485  df-phtpc 18506  df-pcon 23767  df-scon 23768
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