Theorem reparphti 22797

Description: Lemma for reparpht 22798. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

Hypotheses

Ref	Expression
reparpht.2	|- ( ph -> F e. ( II Cn J ) )
reparpht.3	|- ( ph -> G e. ( II Cn II ) )
reparpht.4	|- ( ph -> ( G ` 0 ) = 0 )
reparpht.5	|- ( ph -> ( G ` 1 ) = 1 )
reparphti.6	|- H = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( F ` ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) )

Assertion

Ref	Expression
reparphti	|- ( ph -> H e. ( ( F o. G ) ( PHtpy ` J ) F ) )

Distinct variable groups:   x, y, F   x, G, y   x, J, y   ph, x, y

Allowed substitution hints:   H( x, y)

Proof of Theorem reparphti

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

Step	Hyp	Ref	Expression
1		reparpht.3	. . 3 |- ( ph -> G e. ( II Cn II ) )
2		reparpht.2	. . 3 |- ( ph -> F e. ( II Cn J ) )
3		cnco 21070	. . 3 |- ( ( G e. ( II Cn II ) /\ F e. ( II Cn J ) ) -> ( F o. G ) e. ( II Cn J ) )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> ( F o. G ) e. ( II Cn J ) )
5		reparphti.6	. . 3 |- H = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( F ` ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) )
6		iitopon 22682	. . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
7	6	a1i 11	. . . 4 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
8		eqid 2622	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
9	8	cnfldtop 22587	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) e. Top
10		cnrest2r 21091	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) C_ ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
11	9, 10	mp1i 13	. . . . . . . . 9 |- ( ph -> ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) C_ ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
12	7, 7	cnmpt2nd 21472	. . . . . . . . . . 11 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> y ) e. ( ( II tX II ) Cn II ) )
13		iirevcn 22729	. . . . . . . . . . . 12 |- ( z e. ( 0 [,] 1 ) |-> ( 1 - z ) ) e. ( II Cn II )
14	13	a1i 11	. . . . . . . . . . 11 |- ( ph -> ( z e. ( 0 [,] 1 ) |-> ( 1 - z ) ) e. ( II Cn II ) )
15		oveq2 6658	. . . . . . . . . . 11 |- ( z = y -> ( 1 - z ) = ( 1 - y ) )
16	7, 7, 12, 7, 14, 15	cnmpt21 21474	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( 1 - y ) ) e. ( ( II tX II ) Cn II ) )
17	8	dfii3 22686	. . . . . . . . . . 11 |- II = ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) )
18	17	oveq2i 6661	. . . . . . . . . 10 |- ( ( II tX II ) Cn II ) = ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) )
19	16, 18	syl6eleq 2711	. . . . . . . . 9 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( 1 - y ) ) e. ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) )
20	11, 19	sseldd 3604	. . . . . . . 8 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( 1 - y ) ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
21	7, 7	cnmpt1st 21471	. . . . . . . . . . 11 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> x ) e. ( ( II tX II ) Cn II ) )
22	7, 7, 21, 1	cnmpt21f 21475	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( G ` x ) ) e. ( ( II tX II ) Cn II ) )
23	22, 18	syl6eleq 2711	. . . . . . . . 9 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( G ` x ) ) e. ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) )
24	11, 23	sseldd 3604	. . . . . . . 8 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( G ` x ) ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
25	8	mulcn 22670	. . . . . . . . 9 |- x. e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
26	25	a1i 11	. . . . . . . 8 |- ( ph -> x. e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) ) )
27	7, 7, 20, 24, 26	cnmpt22f 21478	. . . . . . 7 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( 1 - y ) x. ( G ` x ) ) ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
28	12, 18	syl6eleq 2711	. . . . . . . . 9 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> y ) e. ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) )
29	11, 28	sseldd 3604	. . . . . . . 8 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> y ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
30	21, 18	syl6eleq 2711	. . . . . . . . 9 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> x ) e. ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) )
31	11, 30	sseldd 3604	. . . . . . . 8 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> x ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
32	7, 7, 29, 31, 26	cnmpt22f 21478	. . . . . . 7 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( y x. x ) ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
33	8	addcn 22668	. . . . . . . 8 |- + e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) )
34	33	a1i 11	. . . . . . 7 |- ( ph -> + e. ( ( ( TopOpen ` ℂfld ) tX ( TopOpen ` ℂfld ) ) Cn ( TopOpen ` ℂfld ) ) )
35	7, 7, 27, 32, 34	cnmpt22f 21478	. . . . . 6 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) )
36	8	cnfldtopon 22586	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
37	36	a1i 11	. . . . . . 7 |- ( ph -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
38		iiuni 22684	. . . . . . . . . . . . . . 15 |- ( 0 [,] 1 ) = U. II
39	38, 38	cnf 21050	. . . . . . . . . . . . . 14 |- ( G e. ( II Cn II ) -> G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) )
40	1, 39	syl 17	. . . . . . . . . . . . 13 |- ( ph -> G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) )
41	40	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0 [,] 1 ) ) -> ( G ` x ) e. ( 0 [,] 1 ) )
42	41	adantrr 753	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( G ` x ) e. ( 0 [,] 1 ) )
43		simprl 794	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> x e. ( 0 [,] 1 ) )
44		simprr 796	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> y e. ( 0 [,] 1 ) )
45		0re 10040	. . . . . . . . . . . 12 |- 0 e. RR
46		1re 10039	. . . . . . . . . . . 12 |- 1 e. RR
47		icccvx 22749	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ 1 e. RR ) -> ( ( ( G ` x ) e. ( 0 [,] 1 ) /\ x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) -> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) e. ( 0 [,] 1 ) ) )
48	45, 46, 47	mp2an 708	. . . . . . . . . . 11 |- ( ( ( G ` x ) e. ( 0 [,] 1 ) /\ x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) -> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) e. ( 0 [,] 1 ) )
49	42, 43, 44, 48	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) -> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) e. ( 0 [,] 1 ) )
50	49	ralrimivva 2971	. . . . . . . . 9 |- ( ph -> A. x e. ( 0 [,] 1 ) A. y e. ( 0 [,] 1 ) ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) e. ( 0 [,] 1 ) )
51		eqid 2622	. . . . . . . . . 10 |- ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) )
52	51	fmpt2 7237	. . . . . . . . 9 |- ( A. x e. ( 0 [,] 1 ) A. y e. ( 0 [,] 1 ) ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) e. ( 0 [,] 1 ) <-> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> ( 0 [,] 1 ) )
53	50, 52	sylib 208	. . . . . . . 8 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> ( 0 [,] 1 ) )
54		frn 6053	. . . . . . . 8 |- ( ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> ( 0 [,] 1 ) -> ran ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) C_ ( 0 [,] 1 ) )
55	53, 54	syl 17	. . . . . . 7 |- ( ph -> ran ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) C_ ( 0 [,] 1 ) )
56		unitssre 12319	. . . . . . . . 9 |- ( 0 [,] 1 ) C_ RR
57		ax-resscn 9993	. . . . . . . . 9 |- RR C_ CC
58	56, 57	sstri 3612	. . . . . . . 8 |- ( 0 [,] 1 ) C_ CC
59	58	a1i 11	. . . . . . 7 |- ( ph -> ( 0 [,] 1 ) C_ CC )
60		cnrest2 21090	. . . . . . 7 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ran ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) C_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) C_ CC ) -> ( ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) <-> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) e. ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) ) )
61	37, 55, 59, 60	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) e. ( ( II tX II ) Cn ( TopOpen ` ℂfld ) ) <-> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) e. ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) ) )
62	35, 61	mpbid 222	. . . . 5 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) e. ( ( II tX II ) Cn ( ( TopOpen ` ℂfld ) ↾t ( 0 [,] 1 ) ) ) )
63	62, 18	syl6eleqr 2712	. . . 4 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) e. ( ( II tX II ) Cn II ) )
64	7, 7, 63, 2	cnmpt21f 21475	. . 3 |- ( ph -> ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( F ` ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) ) e. ( ( II tX II ) Cn J ) )
65	5, 64	syl5eqel 2705	. 2 |- ( ph -> H e. ( ( II tX II ) Cn J ) )
66	40	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` s ) e. ( 0 [,] 1 ) )
67	58, 66	sseldi 3601	. . . . . . 7 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` s ) e. CC )
68	67	mulid2d 10058	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 x. ( G ` s ) ) = ( G ` s ) )
69	58	sseli 3599	. . . . . . . 8 |- ( s e. ( 0 [,] 1 ) -> s e. CC )
70	69	adantl 482	. . . . . . 7 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> s e. CC )
71	70	mul02d 10234	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 x. s ) = 0 )
72	68, 71	oveq12d 6668	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 x. ( G ` s ) ) + ( 0 x. s ) ) = ( ( G ` s ) + 0 ) )
73	67	addid1d 10236	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( G ` s ) + 0 ) = ( G ` s ) )
74	72, 73	eqtrd 2656	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 x. ( G ` s ) ) + ( 0 x. s ) ) = ( G ` s ) )
75	74	fveq2d 6195	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( ( 1 x. ( G ` s ) ) + ( 0 x. s ) ) ) = ( F ` ( G ` s ) ) )
76		simpr 477	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> s e. ( 0 [,] 1 ) )
77		0elunit 12290	. . . 4 |- 0 e. ( 0 [,] 1 )
78		simpr 477	. . . . . . . . . 10 |- ( ( x = s /\ y = 0 ) -> y = 0 )
79	78	oveq2d 6666	. . . . . . . . 9 |- ( ( x = s /\ y = 0 ) -> ( 1 - y ) = ( 1 - 0 ) )
80		1m0e1 11131	. . . . . . . . 9 |- ( 1 - 0 ) = 1
81	79, 80	syl6eq 2672	. . . . . . . 8 |- ( ( x = s /\ y = 0 ) -> ( 1 - y ) = 1 )
82		simpl 473	. . . . . . . . 9 |- ( ( x = s /\ y = 0 ) -> x = s )
83	82	fveq2d 6195	. . . . . . . 8 |- ( ( x = s /\ y = 0 ) -> ( G ` x ) = ( G ` s ) )
84	81, 83	oveq12d 6668	. . . . . . 7 |- ( ( x = s /\ y = 0 ) -> ( ( 1 - y ) x. ( G ` x ) ) = ( 1 x. ( G ` s ) ) )
85	78, 82	oveq12d 6668	. . . . . . 7 |- ( ( x = s /\ y = 0 ) -> ( y x. x ) = ( 0 x. s ) )
86	84, 85	oveq12d 6668	. . . . . 6 |- ( ( x = s /\ y = 0 ) -> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) = ( ( 1 x. ( G ` s ) ) + ( 0 x. s ) ) )
87	86	fveq2d 6195	. . . . 5 |- ( ( x = s /\ y = 0 ) -> ( F ` ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) = ( F ` ( ( 1 x. ( G ` s ) ) + ( 0 x. s ) ) ) )
88		fvex 6201	. . . . 5 |- ( F ` ( ( 1 x. ( G ` s ) ) + ( 0 x. s ) ) ) e. _V
89	87, 5, 88	ovmpt2a 6791	. . . 4 |- ( ( s e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> ( s H 0 ) = ( F ` ( ( 1 x. ( G ` s ) ) + ( 0 x. s ) ) ) )
90	76, 77, 89	sylancl 694	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s H 0 ) = ( F ` ( ( 1 x. ( G ` s ) ) + ( 0 x. s ) ) ) )
91		fvco3 6275	. . . 4 |- ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. G ) ` s ) = ( F ` ( G ` s ) ) )
92	40, 91	sylan 488	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. G ) ` s ) = ( F ` ( G ` s ) ) )
93	75, 90, 92	3eqtr4d 2666	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s H 0 ) = ( ( F o. G ) ` s ) )
94		1elunit 12291	. . . 4 |- 1 e. ( 0 [,] 1 )
95		simpr 477	. . . . . . . . . 10 |- ( ( x = s /\ y = 1 ) -> y = 1 )
96	95	oveq2d 6666	. . . . . . . . 9 |- ( ( x = s /\ y = 1 ) -> ( 1 - y ) = ( 1 - 1 ) )
97		1m1e0 11089	. . . . . . . . 9 |- ( 1 - 1 ) = 0
98	96, 97	syl6eq 2672	. . . . . . . 8 |- ( ( x = s /\ y = 1 ) -> ( 1 - y ) = 0 )
99		simpl 473	. . . . . . . . 9 |- ( ( x = s /\ y = 1 ) -> x = s )
100	99	fveq2d 6195	. . . . . . . 8 |- ( ( x = s /\ y = 1 ) -> ( G ` x ) = ( G ` s ) )
101	98, 100	oveq12d 6668	. . . . . . 7 |- ( ( x = s /\ y = 1 ) -> ( ( 1 - y ) x. ( G ` x ) ) = ( 0 x. ( G ` s ) ) )
102	95, 99	oveq12d 6668	. . . . . . 7 |- ( ( x = s /\ y = 1 ) -> ( y x. x ) = ( 1 x. s ) )
103	101, 102	oveq12d 6668	. . . . . 6 |- ( ( x = s /\ y = 1 ) -> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) = ( ( 0 x. ( G ` s ) ) + ( 1 x. s ) ) )
104	103	fveq2d 6195	. . . . 5 |- ( ( x = s /\ y = 1 ) -> ( F ` ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) = ( F ` ( ( 0 x. ( G ` s ) ) + ( 1 x. s ) ) ) )
105		fvex 6201	. . . . 5 |- ( F ` ( ( 0 x. ( G ` s ) ) + ( 1 x. s ) ) ) e. _V
106	104, 5, 105	ovmpt2a 6791	. . . 4 |- ( ( s e. ( 0 [,] 1 ) /\ 1 e. ( 0 [,] 1 ) ) -> ( s H 1 ) = ( F ` ( ( 0 x. ( G ` s ) ) + ( 1 x. s ) ) ) )
107	76, 94, 106	sylancl 694	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s H 1 ) = ( F ` ( ( 0 x. ( G ` s ) ) + ( 1 x. s ) ) ) )
108	67	mul02d 10234	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 x. ( G ` s ) ) = 0 )
109	70	mulid2d 10058	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 x. s ) = s )
110	108, 109	oveq12d 6668	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 x. ( G ` s ) ) + ( 1 x. s ) ) = ( 0 + s ) )
111	70	addid2d 10237	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 + s ) = s )
112	110, 111	eqtrd 2656	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 x. ( G ` s ) ) + ( 1 x. s ) ) = s )
113	112	fveq2d 6195	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( ( 0 x. ( G ` s ) ) + ( 1 x. s ) ) ) = ( F ` s ) )
114	107, 113	eqtrd 2656	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s H 1 ) = ( F ` s ) )
115		reparpht.4	. . . . . . . . 9 |- ( ph -> ( G ` 0 ) = 0 )
116	115	adantr 481	. . . . . . . 8 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` 0 ) = 0 )
117	116	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 - s ) x. ( G ` 0 ) ) = ( ( 1 - s ) x. 0 ) )
118		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
119		subcl 10280	. . . . . . . . 9 |- ( ( 1 e. CC /\ s e. CC ) -> ( 1 - s ) e. CC )
120	118, 70, 119	sylancr 695	. . . . . . . 8 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 - s ) e. CC )
121	120	mul01d 10235	. . . . . . 7 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 - s ) x. 0 ) = 0 )
122	117, 121	eqtrd 2656	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 - s ) x. ( G ` 0 ) ) = 0 )
123	70	mul01d 10235	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s x. 0 ) = 0 )
124	122, 123	oveq12d 6668	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( ( 1 - s ) x. ( G ` 0 ) ) + ( s x. 0 ) ) = ( 0 + 0 ) )
125		00id 10211	. . . . 5 |- ( 0 + 0 ) = 0
126	124, 125	syl6eq 2672	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( ( 1 - s ) x. ( G ` 0 ) ) + ( s x. 0 ) ) = 0 )
127	126	fveq2d 6195	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( ( ( 1 - s ) x. ( G ` 0 ) ) + ( s x. 0 ) ) ) = ( F ` 0 ) )
128		simpr 477	. . . . . . . . 9 |- ( ( x = 0 /\ y = s ) -> y = s )
129	128	oveq2d 6666	. . . . . . . 8 |- ( ( x = 0 /\ y = s ) -> ( 1 - y ) = ( 1 - s ) )
130		simpl 473	. . . . . . . . 9 |- ( ( x = 0 /\ y = s ) -> x = 0 )
131	130	fveq2d 6195	. . . . . . . 8 |- ( ( x = 0 /\ y = s ) -> ( G ` x ) = ( G ` 0 ) )
132	129, 131	oveq12d 6668	. . . . . . 7 |- ( ( x = 0 /\ y = s ) -> ( ( 1 - y ) x. ( G ` x ) ) = ( ( 1 - s ) x. ( G ` 0 ) ) )
133	128, 130	oveq12d 6668	. . . . . . 7 |- ( ( x = 0 /\ y = s ) -> ( y x. x ) = ( s x. 0 ) )
134	132, 133	oveq12d 6668	. . . . . 6 |- ( ( x = 0 /\ y = s ) -> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) = ( ( ( 1 - s ) x. ( G ` 0 ) ) + ( s x. 0 ) ) )
135	134	fveq2d 6195	. . . . 5 |- ( ( x = 0 /\ y = s ) -> ( F ` ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) = ( F ` ( ( ( 1 - s ) x. ( G ` 0 ) ) + ( s x. 0 ) ) ) )
136		fvex 6201	. . . . 5 |- ( F ` ( ( ( 1 - s ) x. ( G ` 0 ) ) + ( s x. 0 ) ) ) e. _V
137	135, 5, 136	ovmpt2a 6791	. . . 4 |- ( ( 0 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( 0 H s ) = ( F ` ( ( ( 1 - s ) x. ( G ` 0 ) ) + ( s x. 0 ) ) ) )
138	77, 76, 137	sylancr 695	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 H s ) = ( F ` ( ( ( 1 - s ) x. ( G ` 0 ) ) + ( s x. 0 ) ) ) )
139		fvco3 6275	. . . . . 6 |- ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> ( ( F o. G ) ` 0 ) = ( F ` ( G ` 0 ) ) )
140	40, 77, 139	sylancl 694	. . . . 5 |- ( ph -> ( ( F o. G ) ` 0 ) = ( F ` ( G ` 0 ) ) )
141	115	fveq2d 6195	. . . . 5 |- ( ph -> ( F ` ( G ` 0 ) ) = ( F ` 0 ) )
142	140, 141	eqtrd 2656	. . . 4 |- ( ph -> ( ( F o. G ) ` 0 ) = ( F ` 0 ) )
143	142	adantr 481	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. G ) ` 0 ) = ( F ` 0 ) )
144	127, 138, 143	3eqtr4d 2666	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 H s ) = ( ( F o. G ) ` 0 ) )
145		reparpht.5	. . . . . . . . 9 |- ( ph -> ( G ` 1 ) = 1 )
146	145	adantr 481	. . . . . . . 8 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` 1 ) = 1 )
147	146	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 - s ) x. ( G ` 1 ) ) = ( ( 1 - s ) x. 1 ) )
148	120	mulid1d 10057	. . . . . . 7 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 - s ) x. 1 ) = ( 1 - s ) )
149	147, 148	eqtrd 2656	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 - s ) x. ( G ` 1 ) ) = ( 1 - s ) )
150	70	mulid1d 10057	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s x. 1 ) = s )
151	149, 150	oveq12d 6668	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( ( 1 - s ) x. ( G ` 1 ) ) + ( s x. 1 ) ) = ( ( 1 - s ) + s ) )
152		npcan 10290	. . . . . 6 |- ( ( 1 e. CC /\ s e. CC ) -> ( ( 1 - s ) + s ) = 1 )
153	118, 70, 152	sylancr 695	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 1 - s ) + s ) = 1 )
154	151, 153	eqtrd 2656	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( ( 1 - s ) x. ( G ` 1 ) ) + ( s x. 1 ) ) = 1 )
155	154	fveq2d 6195	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( ( ( 1 - s ) x. ( G ` 1 ) ) + ( s x. 1 ) ) ) = ( F ` 1 ) )
156		simpr 477	. . . . . . . . 9 |- ( ( x = 1 /\ y = s ) -> y = s )
157	156	oveq2d 6666	. . . . . . . 8 |- ( ( x = 1 /\ y = s ) -> ( 1 - y ) = ( 1 - s ) )
158		simpl 473	. . . . . . . . 9 |- ( ( x = 1 /\ y = s ) -> x = 1 )
159	158	fveq2d 6195	. . . . . . . 8 |- ( ( x = 1 /\ y = s ) -> ( G ` x ) = ( G ` 1 ) )
160	157, 159	oveq12d 6668	. . . . . . 7 |- ( ( x = 1 /\ y = s ) -> ( ( 1 - y ) x. ( G ` x ) ) = ( ( 1 - s ) x. ( G ` 1 ) ) )
161	156, 158	oveq12d 6668	. . . . . . 7 |- ( ( x = 1 /\ y = s ) -> ( y x. x ) = ( s x. 1 ) )
162	160, 161	oveq12d 6668	. . . . . 6 |- ( ( x = 1 /\ y = s ) -> ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) = ( ( ( 1 - s ) x. ( G ` 1 ) ) + ( s x. 1 ) ) )
163	162	fveq2d 6195	. . . . 5 |- ( ( x = 1 /\ y = s ) -> ( F ` ( ( ( 1 - y ) x. ( G ` x ) ) + ( y x. x ) ) ) = ( F ` ( ( ( 1 - s ) x. ( G ` 1 ) ) + ( s x. 1 ) ) ) )
164		fvex 6201	. . . . 5 |- ( F ` ( ( ( 1 - s ) x. ( G ` 1 ) ) + ( s x. 1 ) ) ) e. _V
165	163, 5, 164	ovmpt2a 6791	. . . 4 |- ( ( 1 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( 1 H s ) = ( F ` ( ( ( 1 - s ) x. ( G ` 1 ) ) + ( s x. 1 ) ) ) )
166	94, 76, 165	sylancr 695	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 H s ) = ( F ` ( ( ( 1 - s ) x. ( G ` 1 ) ) + ( s x. 1 ) ) ) )
167		fvco3 6275	. . . . . 6 |- ( ( G : ( 0 [,] 1 ) --> ( 0 [,] 1 ) /\ 1 e. ( 0 [,] 1 ) ) -> ( ( F o. G ) ` 1 ) = ( F ` ( G ` 1 ) ) )
168	40, 94, 167	sylancl 694	. . . . 5 |- ( ph -> ( ( F o. G ) ` 1 ) = ( F ` ( G ` 1 ) ) )
169	145	fveq2d 6195	. . . . 5 |- ( ph -> ( F ` ( G ` 1 ) ) = ( F ` 1 ) )
170	168, 169	eqtrd 2656	. . . 4 |- ( ph -> ( ( F o. G ) ` 1 ) = ( F ` 1 ) )
171	170	adantr 481	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. G ) ` 1 ) = ( F ` 1 ) )
172	155, 166, 171	3eqtr4d 2666	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 H s ) = ( ( F o. G ) ` 1 ) )
173	4, 2, 65, 93, 114, 144, 172	isphtpy2d 22786	1 |- ( ph -> H e. ( ( F o. G ) ( PHtpy ` J ) F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   |-> cmpt 4729   X. cxp 5112   ran crn 5115   o. ccom 5118   -->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

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-ioo 12179  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

This theorem is referenced by:  reparpht  22798