Theorem cvxsconn 31225

Description: A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.)

Hypotheses

Ref	Expression
cvxpconn.1	|- ( ph -> S C_ CC )
cvxpconn.2	|- ( ( ph /\ ( x e. S /\ y e. S /\ t e. ( 0 [,] 1 ) ) ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. S )
cvxpconn.3	|- J = ( TopOpen ` ℂfld )
cvxpconn.4	|- K = ( J ↾t S )

Assertion

Ref	Expression
cvxsconn	|- ( ph -> K e. SConn )

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 cvxsconn

Dummy variables z f s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvxpconn.1	. . 3 |- ( ph -> S C_ CC )
2		cvxpconn.2	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ t e. ( 0 [,] 1 ) ) ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. S )
3		cvxpconn.3	. . 3 |- J = ( TopOpen ` ℂfld )
4		cvxpconn.4	. . 3 |- K = ( J ↾t S )
5	1, 2, 3, 4	cvxpconn 31224	. 2 |- ( ph -> K e. PConn )
6		simprl 794	. . . . 5 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> f e. ( II Cn K ) )
7		pconntop 31207	. . . . . . . . . 10 |- ( K e. PConn -> K e. Top )
8	5, 7	syl 17	. . . . . . . . 9 |- ( ph -> K e. Top )
9	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> K e. Top )
10		eqid 2622	. . . . . . . . 9 |- U. K = U. K
11	10	toptopon 20722	. . . . . . . 8 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
12	9, 11	sylib 208	. . . . . . 7 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> K e. (TopOn ` U. K ) )
13		iiuni 22684	. . . . . . . . . 10 |- ( 0 [,] 1 ) = U. II
14	13, 10	cnf 21050	. . . . . . . . 9 |- ( f e. ( II Cn K ) -> f : ( 0 [,] 1 ) --> U. K )
15	6, 14	syl 17	. . . . . . . 8 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> f : ( 0 [,] 1 ) --> U. K )
16		0elunit 12290	. . . . . . . 8 |- 0 e. ( 0 [,] 1 )
17		ffvelrn 6357	. . . . . . . 8 |- ( ( f : ( 0 [,] 1 ) --> U. K /\ 0 e. ( 0 [,] 1 ) ) -> ( f ` 0 ) e. U. K )
18	15, 16, 17	sylancl 694	. . . . . . 7 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( f ` 0 ) e. U. K )
19		eqid 2622	. . . . . . . 8 |- ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) = ( ( 0 [,] 1 ) X. { ( f ` 0 ) } )
20	19	pcoptcl 22821	. . . . . . 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 ) ) )
21	12, 18, 20	syl2anc 693	. . . . . 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 ) ) )
22	21	simp1d 1073	. . . . 5 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) e. ( II Cn K ) )
23		iitopon 22682	. . . . . . . . . . 11 |- II e. (TopOn ` ( 0 [,] 1 ) )
24	23	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> II e. (TopOn ` ( 0 [,] 1 ) ) )
25	3	dfii3 22686	. . . . . . . . . . . 12 |- II = ( J ↾t ( 0 [,] 1 ) )
26	3	cnfldtopon 22586	. . . . . . . . . . . . 13 |- J e. (TopOn ` CC )
27	26	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> J e. (TopOn ` CC ) )
28		unitssre 12319	. . . . . . . . . . . . . 14 |- ( 0 [,] 1 ) C_ RR
29		ax-resscn 9993	. . . . . . . . . . . . . 14 |- RR C_ CC
30	28, 29	sstri 3612	. . . . . . . . . . . . 13 |- ( 0 [,] 1 ) C_ CC
31	30	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( 0 [,] 1 ) C_ CC )
32	27, 27	cnmpt2nd 21472	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( z e. CC , t e. CC |-> t ) e. ( ( J tX J ) Cn J ) )
33	25, 27, 31, 25, 27, 31, 32	cnmpt2res 21480	. . . . . . . . . . 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 ) )
34	1	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> S C_ CC )
35		resttopon 20965	. . . . . . . . . . . . . . . . . 18 |- ( ( J e. (TopOn ` CC ) /\ S C_ CC ) -> ( J ↾t S ) e. (TopOn ` S ) )
36	26, 1, 35	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( J ↾t S ) e. (TopOn ` S ) )
37	4, 36	syl5eqel 2705	. . . . . . . . . . . . . . . 16 |- ( ph -> K e. (TopOn ` S ) )
38		toponuni 20719	. . . . . . . . . . . . . . . 16 |- ( K e. (TopOn ` S ) -> S = U. K )
39	37, 38	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> S = U. K )
40	39	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> S = U. K )
41	18, 40	eleqtrrd 2704	. . . . . . . . . . . . 13 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( f ` 0 ) e. S )
42	34, 41	sseldd 3604	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( f ` 0 ) e. CC )
43	24, 24, 27, 42	cnmpt2c 21473	. . . . . . . . . . 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 ) )
44	3	mulcn 22670	. . . . . . . . . . . 12 |- x. e. ( ( J tX J ) Cn J )
45	44	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> x. e. ( ( J tX J ) Cn J ) )
46	24, 24, 33, 43, 45	cnmpt22f 21478	. . . . . . . . . 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 9994	. . . . . . . . . . . . . . 15 |- 1 e. CC
48	47	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> 1 e. CC )
49	27, 27, 27, 48	cnmpt2c 21473	. . . . . . . . . . . . 13 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( z e. CC , t e. CC |-> 1 ) e. ( ( J tX J ) Cn J ) )
50	3	subcn 22669	. . . . . . . . . . . . . 14 |- - e. ( ( J tX J ) Cn J )
51	50	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> - e. ( ( J tX J ) Cn J ) )
52	27, 27, 49, 32, 51	cnmpt22f 21478	. . . . . . . . . . . 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 ) )
53	25, 27, 31, 25, 27, 31, 52	cnmpt2res 21480	. . . . . . . . . . 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 ) )
54	24, 24	cnmpt1st 21471	. . . . . . . . . . . 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 ) )
55	3	cnfldtop 22587	. . . . . . . . . . . . . 14 |- J e. Top
56		cnrest2r 21091	. . . . . . . . . . . . . 14 |- ( J e. Top -> ( II Cn ( J ↾t S ) ) C_ ( II Cn J ) )
57	55, 56	ax-mp 5	. . . . . . . . . . . . 13 |- ( II Cn ( J ↾t S ) ) C_ ( II Cn J )
58	4	oveq2i 6661	. . . . . . . . . . . . . 14 |- ( II Cn K ) = ( II Cn ( J ↾t S ) )
59	6, 58	syl6eleq 2711	. . . . . . . . . . . . 13 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> f e. ( II Cn ( J ↾t S ) ) )
60	57, 59	sseldi 3601	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> f e. ( II Cn J ) )
61	24, 24, 54, 60	cnmpt21f 21475	. . . . . . . . . . 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 ) )
62	24, 24, 53, 61, 45	cnmpt22f 21478	. . . . . . . . . 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 ) )
63	3	addcn 22668	. . . . . . . . . . 11 |- + e. ( ( J tX J ) Cn J )
64	63	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> + e. ( ( J tX J ) Cn J ) )
65	24, 24, 46, 62, 64	cnmpt22f 21478	. . . . . . . . 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 ) )
66	41	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ ( z e. ( 0 [,] 1 ) /\ t e. ( 0 [,] 1 ) ) ) -> ( f ` 0 ) e. S )
67	15	adantr 481	. . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ ( z e. ( 0 [,] 1 ) /\ t e. ( 0 [,] 1 ) ) ) -> z e. ( 0 [,] 1 ) )
69	67, 68	ffvelrnd 6360	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ ( z e. ( 0 [,] 1 ) /\ t e. ( 0 [,] 1 ) ) ) -> ( f ` z ) e. U. K )
70	40	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ ( z e. ( 0 [,] 1 ) /\ t e. ( 0 [,] 1 ) ) ) -> S = U. K )
71	69, 70	eleqtrrd 2704	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ ( z e. ( 0 [,] 1 ) /\ t e. ( 0 [,] 1 ) ) ) -> ( f ` z ) e. S )
72	2	3exp2 1285	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( x e. S -> ( y e. S -> ( t e. ( 0 [,] 1 ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. S ) ) ) )
73	72	imp42 620	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( x e. S /\ y e. S ) ) /\ t e. ( 0 [,] 1 ) ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. S )
74	73	an32s 846	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ t e. ( 0 [,] 1 ) ) /\ ( x e. S /\ y e. S ) ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. S )
75	74	ralrimivva 2971	. . . . . . . . . . . . . . 15 |- ( ( ph /\ t e. ( 0 [,] 1 ) ) -> A. x e. S A. y e. S ( ( t x. x ) + ( ( 1 - t ) x. y ) ) e. S )
76	75	ad2ant2rl 785	. . . . . . . . . . . . . 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 6658	. . . . . . . . . . . . . . . . 17 |- ( x = ( f ` 0 ) -> ( t x. x ) = ( t x. ( f ` 0 ) ) )
78	77	oveq1d 6665	. . . . . . . . . . . . . . . 16 |- ( x = ( f ` 0 ) -> ( ( t x. x ) + ( ( 1 - t ) x. y ) ) = ( ( t x. ( f ` 0 ) ) + ( ( 1 - t ) x. y ) ) )
79	78	eleq1d 2686	. . . . . . . . . . . . . . 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 6658	. . . . . . . . . . . . . . . . 17 |- ( y = ( f ` z ) -> ( ( 1 - t ) x. y ) = ( ( 1 - t ) x. ( f ` z ) ) )
81	80	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( y = ( f ` z ) -> ( ( t x. ( f ` 0 ) ) + ( ( 1 - t ) x. y ) ) = ( ( t x. ( f ` 0 ) ) + ( ( 1 - t ) x. ( f ` z ) ) ) )
82	81	eleq1d 2686	. . . . . . . . . . . . . . 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 ) )
83	79, 82	rspc2va 3323	. . . . . . . . . . . . . 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 )
84	66, 71, 76, 83	syl21anc 1325	. . . . . . . . . . . . 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 )
85	84	ralrimivva 2971	. . . . . . . . . . . 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 2622	. . . . . . . . . . . . 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 ) ) ) )
87	86	fmpt2 7237	. . . . . . . . . . . 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 )
88	85, 87	sylib 208	. . . . . . . . . . 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 6053	. . . . . . . . . . 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 )
90	88, 89	syl 17	. . . . . . . . . 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 21090	. . . . . . . . . 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 ( J ↾t S ) ) ) )
92	27, 90, 34, 91	syl3anc 1326	. . . . . . . . 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 ( J ↾t S ) ) ) )
93	65, 92	mpbid 222	. . . . . . . 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 ( J ↾t S ) ) )
94	4	oveq2i 6661	. . . . . . . 8 |- ( ( II tX II ) Cn K ) = ( ( II tX II ) Cn ( J ↾t S ) )
95	93, 94	syl6eleqr 2712	. . . . . . 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 477	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> s e. ( 0 [,] 1 ) )
97		simpr 477	. . . . . . . . . . . 12 |- ( ( z = s /\ t = 0 ) -> t = 0 )
98	97	oveq1d 6665	. . . . . . . . . . 11 |- ( ( z = s /\ t = 0 ) -> ( t x. ( f ` 0 ) ) = ( 0 x. ( f ` 0 ) ) )
99	97	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( z = s /\ t = 0 ) -> ( 1 - t ) = ( 1 - 0 ) )
100		1m0e1 11131	. . . . . . . . . . . . 13 |- ( 1 - 0 ) = 1
101	99, 100	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( z = s /\ t = 0 ) -> ( 1 - t ) = 1 )
102		simpl 473	. . . . . . . . . . . . 13 |- ( ( z = s /\ t = 0 ) -> z = s )
103	102	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( z = s /\ t = 0 ) -> ( f ` z ) = ( f ` s ) )
104	101, 103	oveq12d 6668	. . . . . . . . . . 11 |- ( ( z = s /\ t = 0 ) -> ( ( 1 - t ) x. ( f ` z ) ) = ( 1 x. ( f ` s ) ) )
105	98, 104	oveq12d 6668	. . . . . . . . . 10 |- ( ( z = s /\ t = 0 ) -> ( ( t x. ( f ` 0 ) ) + ( ( 1 - t ) x. ( f ` z ) ) ) = ( ( 0 x. ( f ` 0 ) ) + ( 1 x. ( f ` s ) ) ) )
106		ovex 6678	. . . . . . . . . 10 |- ( ( 0 x. ( f ` 0 ) ) + ( 1 x. ( f ` s ) ) ) e. _V
107	105, 86, 106	ovmpt2a 6791	. . . . . . . . 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 ) ) ) )
108	96, 16, 107	sylancl 694	. . . . . . . 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 ) ) ) )
109	42	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( f ` 0 ) e. CC )
110	109	mul02d 10234	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( 0 x. ( f ` 0 ) ) = 0 )
111	26	toponunii 20721	. . . . . . . . . . . . 13 |- CC = U. J
112	13, 111	cnf 21050	. . . . . . . . . . . 12 |- ( f e. ( II Cn J ) -> f : ( 0 [,] 1 ) --> CC )
113	60, 112	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> f : ( 0 [,] 1 ) --> CC )
114	113	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( f ` s ) e. CC )
115	114	mulid2d 10058	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( 1 x. ( f ` s ) ) = ( f ` s ) )
116	110, 115	oveq12d 6668	. . . . . . . 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 ) ) )
117	114	addid2d 10237	. . . . . . . 8 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( 0 + ( f ` s ) ) = ( f ` s ) )
118	108, 116, 117	3eqtrd 2660	. . . . . . 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 12291	. . . . . . . . 9 |- 1 e. ( 0 [,] 1 )
120		simpr 477	. . . . . . . . . . . 12 |- ( ( z = s /\ t = 1 ) -> t = 1 )
121	120	oveq1d 6665	. . . . . . . . . . 11 |- ( ( z = s /\ t = 1 ) -> ( t x. ( f ` 0 ) ) = ( 1 x. ( f ` 0 ) ) )
122	120	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( z = s /\ t = 1 ) -> ( 1 - t ) = ( 1 - 1 ) )
123		1m1e0 11089	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
124	122, 123	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( z = s /\ t = 1 ) -> ( 1 - t ) = 0 )
125		simpl 473	. . . . . . . . . . . . 13 |- ( ( z = s /\ t = 1 ) -> z = s )
126	125	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( z = s /\ t = 1 ) -> ( f ` z ) = ( f ` s ) )
127	124, 126	oveq12d 6668	. . . . . . . . . . 11 |- ( ( z = s /\ t = 1 ) -> ( ( 1 - t ) x. ( f ` z ) ) = ( 0 x. ( f ` s ) ) )
128	121, 127	oveq12d 6668	. . . . . . . . . 10 |- ( ( z = s /\ t = 1 ) -> ( ( t x. ( f ` 0 ) ) + ( ( 1 - t ) x. ( f ` z ) ) ) = ( ( 1 x. ( f ` 0 ) ) + ( 0 x. ( f ` s ) ) ) )
129		ovex 6678	. . . . . . . . . 10 |- ( ( 1 x. ( f ` 0 ) ) + ( 0 x. ( f ` s ) ) ) e. _V
130	128, 86, 129	ovmpt2a 6791	. . . . . . . . 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 ) ) ) )
131	96, 119, 130	sylancl 694	. . . . . . . 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 ) ) ) )
132	109	mulid2d 10058	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( 1 x. ( f ` 0 ) ) = ( f ` 0 ) )
133	114	mul02d 10234	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( 0 x. ( f ` s ) ) = 0 )
134	132, 133	oveq12d 6668	. . . . . . . 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 ) )
135	109	addid1d 10236	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( ( f ` 0 ) + 0 ) = ( f ` 0 ) )
136		fvex 6201	. . . . . . . . . . 11 |- ( f ` 0 ) e. _V
137	136	fvconst2 6469	. . . . . . . . . 10 |- ( s e. ( 0 [,] 1 ) -> ( ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ` s ) = ( f ` 0 ) )
138	137	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ` s ) = ( f ` 0 ) )
139	135, 138	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( ( f ` 0 ) + 0 ) = ( ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ` s ) )
140	131, 134, 139	3eqtrd 2660	. . . . . . 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 477	. . . . . . . . . . . 12 |- ( ( z = 0 /\ t = s ) -> t = s )
142	141	oveq1d 6665	. . . . . . . . . . 11 |- ( ( z = 0 /\ t = s ) -> ( t x. ( f ` 0 ) ) = ( s x. ( f ` 0 ) ) )
143	141	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( z = 0 /\ t = s ) -> ( 1 - t ) = ( 1 - s ) )
144		simpl 473	. . . . . . . . . . . . 13 |- ( ( z = 0 /\ t = s ) -> z = 0 )
145	144	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( z = 0 /\ t = s ) -> ( f ` z ) = ( f ` 0 ) )
146	143, 145	oveq12d 6668	. . . . . . . . . . 11 |- ( ( z = 0 /\ t = s ) -> ( ( 1 - t ) x. ( f ` z ) ) = ( ( 1 - s ) x. ( f ` 0 ) ) )
147	142, 146	oveq12d 6668	. . . . . . . . . 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 6678	. . . . . . . . . 10 |- ( ( s x. ( f ` 0 ) ) + ( ( 1 - s ) x. ( f ` 0 ) ) ) e. _V
149	147, 86, 148	ovmpt2a 6791	. . . . . . . . 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 ) ) ) )
150	16, 96, 149	sylancr 695	. . . . . . . 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 ) ) ) )
151	30, 96	sseldi 3601	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> s e. CC )
152		pncan3 10289	. . . . . . . . . . 11 |- ( ( s e. CC /\ 1 e. CC ) -> ( s + ( 1 - s ) ) = 1 )
153	151, 47, 152	sylancl 694	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( s + ( 1 - s ) ) = 1 )
154	153	oveq1d 6665	. . . . . . . . 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 10280	. . . . . . . . . . 11 |- ( ( 1 e. CC /\ s e. CC ) -> ( 1 - s ) e. CC )
156	47, 151, 155	sylancr 695	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( 1 - s ) e. CC )
157	151, 156, 109	adddird 10065	. . . . . . . . 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 ) ) ) )
158	154, 157, 132	3eqtr3d 2664	. . . . . . . 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 ) )
159	150, 158	eqtrd 2656	. . . . . . 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 477	. . . . . . . . . . . 12 |- ( ( z = 1 /\ t = s ) -> t = s )
161	160	oveq1d 6665	. . . . . . . . . . 11 |- ( ( z = 1 /\ t = s ) -> ( t x. ( f ` 0 ) ) = ( s x. ( f ` 0 ) ) )
162	160	oveq2d 6666	. . . . . . . . . . . 12 |- ( ( z = 1 /\ t = s ) -> ( 1 - t ) = ( 1 - s ) )
163		simpl 473	. . . . . . . . . . . . 13 |- ( ( z = 1 /\ t = s ) -> z = 1 )
164	163	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( z = 1 /\ t = s ) -> ( f ` z ) = ( f ` 1 ) )
165	162, 164	oveq12d 6668	. . . . . . . . . . 11 |- ( ( z = 1 /\ t = s ) -> ( ( 1 - t ) x. ( f ` z ) ) = ( ( 1 - s ) x. ( f ` 1 ) ) )
166	161, 165	oveq12d 6668	. . . . . . . . . 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 6678	. . . . . . . . . 10 |- ( ( s x. ( f ` 0 ) ) + ( ( 1 - s ) x. ( f ` 1 ) ) ) e. _V
168	166, 86, 167	ovmpt2a 6791	. . . . . . . . 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 ) ) ) )
169	119, 96, 168	sylancr 695	. . . . . . . 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 801	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) /\ s e. ( 0 [,] 1 ) ) -> ( f ` 0 ) = ( f ` 1 ) )
171	170	oveq2d 6666	. . . . . . . . . 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 ) ) )
172	171	oveq2d 6666	. . . . . . . . 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 ) ) ) )
173	158, 172, 170	3eqtr3d 2664	. . . . . . . 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 ) )
174	169, 173	eqtrd 2656	. . . . . . 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 ) )
175	6, 22, 95, 118, 140, 159, 174	isphtpy2d 22786	. . . . . 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 3921	. . . . . 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 ) } ) ) =/= (/) )
177	175, 176	syl 17	. . . . 5 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> ( f ( PHtpy ` K ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) =/= (/) )
178		isphtpc 22793	. . . . 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 ) } ) ) =/= (/) ) )
179	6, 22, 177, 178	syl3anbrc 1246	. . . 4 |- ( ( ph /\ ( f e. ( II Cn K ) /\ ( f ` 0 ) = ( f ` 1 ) ) ) -> f ( ~=ph ` K ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) )
180	179	expr 643	. . 3 |- ( ( ph /\ f e. ( II Cn K ) ) -> ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` K ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) )
181	180	ralrimiva 2966	. 2 |- ( ph -> A. f e. ( II Cn K ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` K ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) )
182		issconn 31208	. 2 |- ( K e. SConn <-> ( K e. PConn /\ A. f e. ( II Cn K ) ( ( f ` 0 ) = ( f ` 1 ) -> f ( ~=ph ` K ) ( ( 0 [,] 1 ) X. { ( f ` 0 ) } ) ) ) )
183	5, 181, 182	sylanbrc 698	1 |- ( ph -> K e. SConn )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   class class class wbr 4653   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   [,]cicc 12178   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   IIcii 22678   PHtpycphtpy 22767   ~=ph cphtpc 22768  PConncpconn 31201  SConncsconn 31202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-pconn 31203  df-sconn 31204

This theorem is referenced by:  blsconn  31226  resconn  31228