Theorem cvmliftphtlem 27206

Description: Lemma for cvmliftpht 27207. (Contributed by Mario Carneiro, 6-Jul-2015.)

Hypotheses

Ref	Expression
cvmliftpht.b	|- B = U. C
cvmliftpht.m	|- M = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
cvmliftpht.n	|- N = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
cvmliftpht.f	|- ( ph -> F e. ( C CovMap J ) )
cvmliftpht.p	|- ( ph -> P e. B )
cvmliftpht.e	|- ( ph -> ( F ` P ) = ( G ` 0 ) )
cvmliftphtlem.g	|- ( ph -> G e. ( II Cn J ) )
cvmliftphtlem.h	|- ( ph -> H e. ( II Cn J ) )
cvmliftphtlem.k	|- ( ph -> K e. ( G ( PHtpy ` J ) H ) )
cvmliftphtlem.a	|- ( ph -> A e. ( ( II tX II ) Cn C ) )
cvmliftphtlem.c	|- ( ph -> ( F o. A ) = K )
cvmliftphtlem.0	|- ( ph -> ( 0 A 0 ) = P )

Assertion

Ref	Expression
cvmliftphtlem	|- ( ph -> A e. ( M ( PHtpy ` C ) N ) )

Distinct variable groups:   A, f   B, f   f, F   f, J   C, f   f, G   f, H   P, f

Allowed substitution hints:   ph( f)   K( f)   M( f)   N( f)

Proof of Theorem cvmliftphtlem

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

Step	Hyp	Ref	Expression
1		cvmliftpht.b	. . . 4 |- B = U. C
2		cvmliftpht.m	. . . 4 |- M = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
3		cvmliftpht.f	. . . 4 |- ( ph -> F e. ( C CovMap J ) )
4		cvmliftphtlem.g	. . . 4 |- ( ph -> G e. ( II Cn J ) )
5		cvmliftpht.p	. . . 4 |- ( ph -> P e. B )
6		cvmliftpht.e	. . . 4 |- ( ph -> ( F ` P ) = ( G ` 0 ) )
7	1, 2, 3, 4, 5, 6	cvmliftiota 27190	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
8	7	simp1d 1000	. 2 |- ( ph -> M e. ( II Cn C ) )
9		cvmliftpht.n	. . . 4 |- N = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
10		cvmliftphtlem.h	. . . 4 |- ( ph -> H e. ( II Cn J ) )
11		cvmliftphtlem.k	. . . . . . 7 |- ( ph -> K e. ( G ( PHtpy ` J ) H ) )
12	4, 10, 11	phtpy01 20557	. . . . . 6 |- ( ph -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
13	12	simpld 459	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
14	6, 13	eqtrd 2475	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
15	1, 9, 3, 10, 5, 14	cvmliftiota 27190	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
16	15	simp1d 1000	. 2 |- ( ph -> N e. ( II Cn C ) )
17		cvmliftphtlem.a	. 2 |- ( ph -> A e. ( ( II tX II ) Cn C ) )
18		iitop 20456	. . . . . . . . . . . . . . . 16 |- II e. Top
19		iiuni 20457	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) = U. II
20	18, 18, 19, 19	txunii 19166	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
21	20, 1	cnf 18850	. . . . . . . . . . . . . 14 |- ( A e. ( ( II tX II ) Cn C ) -> A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B )
22	17, 21	syl 16	. . . . . . . . . . . . 13 |- ( ph -> A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B )
23		0elunit 11403	. . . . . . . . . . . . . 14 |- 0 e. ( 0 [,] 1 )
24		opelxpi 4871	. . . . . . . . . . . . . 14 |- ( ( s e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> <. s , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
25	23, 24	mpan2 671	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. s , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
26		fvco3 5768	. . . . . . . . . . . . 13 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ <. s , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( F o. A ) ` <. s , 0 >. ) = ( F ` ( A ` <. s , 0 >. ) ) )
27	22, 25, 26	syl2an 477	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 0 >. ) = ( F ` ( A ` <. s , 0 >. ) ) )
28		cvmliftphtlem.c	. . . . . . . . . . . . . 14 |- ( ph -> ( F o. A ) = K )
29	28	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. A ) = K )
30	29	fveq1d 5693	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 0 >. ) = ( K ` <. s , 0 >. ) )
31	27, 30	eqtr3d 2477	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 0 >. ) ) = ( K ` <. s , 0 >. ) )
32		df-ov 6094	. . . . . . . . . . . 12 |- ( s A 0 ) = ( A ` <. s , 0 >. )
33	32	fveq2i 5694	. . . . . . . . . . 11 |- ( F ` ( s A 0 ) ) = ( F ` ( A ` <. s , 0 >. ) )
34		df-ov 6094	. . . . . . . . . . 11 |- ( s K 0 ) = ( K ` <. s , 0 >. )
35	31, 33, 34	3eqtr4g 2500	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( s K 0 ) )
36		iitopon 20455	. . . . . . . . . . . . 13 |- II e. (TopOn ` ( 0 [,] 1 ) )
37	36	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
38	4, 10	phtpyhtpy 20554	. . . . . . . . . . . . 13 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( G ( II Htpy J ) H ) )
39	38, 11	sseldd 3357	. . . . . . . . . . . 12 |- ( ph -> K e. ( G ( II Htpy J ) H ) )
40	37, 4, 10, 39	htpyi 20546	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( s K 0 ) = ( G ` s ) /\ ( s K 1 ) = ( H ` s ) ) )
41	40	simpld 459	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s K 0 ) = ( G ` s ) )
42	35, 41	eqtrd 2475	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( G ` s ) )
43	42	mpteq2dva 4378	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
44		fovrn 6233	. . . . . . . . . . 11 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
45	23, 44	mp3an3 1303	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
46	22, 45	sylan 471	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
47		eqidd 2444	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
48		cvmcn 27151	. . . . . . . . . . . 12 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
49	3, 48	syl 16	. . . . . . . . . . 11 |- ( ph -> F e. ( C Cn J ) )
50		eqid 2443	. . . . . . . . . . . 12 |- U. J = U. J
51	1, 50	cnf 18850	. . . . . . . . . . 11 |- ( F e. ( C Cn J ) -> F : B --> U. J )
52	49, 51	syl 16	. . . . . . . . . 10 |- ( ph -> F : B --> U. J )
53	52	feqmptd 5744	. . . . . . . . 9 |- ( ph -> F = ( x e. B |-> ( F ` x ) ) )
54		fveq2 5691	. . . . . . . . 9 |- ( x = ( s A 0 ) -> ( F ` x ) = ( F ` ( s A 0 ) ) )
55	46, 47, 53, 54	fmptco 5876	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) )
56	19, 50	cnf 18850	. . . . . . . . . 10 |- ( G e. ( II Cn J ) -> G : ( 0 [,] 1 ) --> U. J )
57	4, 56	syl 16	. . . . . . . . 9 |- ( ph -> G : ( 0 [,] 1 ) --> U. J )
58	57	feqmptd 5744	. . . . . . . 8 |- ( ph -> G = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
59	43, 55, 58	3eqtr4d 2485	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G )
60		cvmliftphtlem.0	. . . . . . 7 |- ( ph -> ( 0 A 0 ) = P )
61	37	cnmptid 19234	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> s ) e. ( II Cn II ) )
62	23	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 e. ( 0 [,] 1 ) )
63	37, 37, 62	cnmptc 19235	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
64	37, 61, 63, 17	cnmpt12f 19239	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) e. ( II Cn C ) )
65	1	cvmlift 27188	. . . . . . . . 9 |- ( ( ( F e. ( C CovMap J ) /\ G e. ( II Cn J ) ) /\ ( P e. B /\ ( F ` P ) = ( G ` 0 ) ) ) -> E! f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
66	3, 4, 5, 6, 65	syl22anc 1219	. . . . . . . 8 |- ( ph -> E! f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
67		coeq2 4998	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( F o. f ) = ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) )
68	67	eqeq1d 2451	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( ( F o. f ) = G <-> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G ) )
69		fveq1 5690	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) )
70		oveq1 6098	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 0 ) = ( 0 A 0 ) )
71		eqid 2443	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) )
72		ovex 6116	. . . . . . . . . . . . . 14 |- ( 0 A 0 ) e. _V
73	70, 71, 72	fvmpt 5774	. . . . . . . . . . . . 13 |- ( 0 e. ( 0 [,] 1 ) -> ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) = ( 0 A 0 ) )
74	23, 73	ax-mp 5	. . . . . . . . . . . 12 |- ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) = ( 0 A 0 )
75	69, 74	syl6eq 2491	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( 0 A 0 ) )
76	75	eqeq1d 2451	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( ( f ` 0 ) = P <-> ( 0 A 0 ) = P ) )
77	68, 76	anbi12d 710	. . . . . . . . 9 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( ( ( F o. f ) = G /\ ( f ` 0 ) = P ) <-> ( ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G /\ ( 0 A 0 ) = P ) ) )
78	77	riota2 6075	. . . . . . . 8 |- ( ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) e. ( II Cn C ) /\ E! f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) ) -> ( ( ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G /\ ( 0 A 0 ) = P ) <-> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) )
79	64, 66, 78	syl2anc 661	. . . . . . 7 |- ( ph -> ( ( ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G /\ ( 0 A 0 ) = P ) <-> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) )
80	59, 60, 79	mpbi2and 912	. . . . . 6 |- ( ph -> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
81	2, 80	syl5eq 2487	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
82	19, 1	cnf 18850	. . . . . . 7 |- ( M e. ( II Cn C ) -> M : ( 0 [,] 1 ) --> B )
83	8, 82	syl 16	. . . . . 6 |- ( ph -> M : ( 0 [,] 1 ) --> B )
84	83	feqmptd 5744	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
85	81, 84	eqtr3d 2477	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
86		mpteqb 5788	. . . . 5 |- ( A. s e. ( 0 [,] 1 ) ( s A 0 ) e. _V -> ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) <-> A. s e. ( 0 [,] 1 ) ( s A 0 ) = ( M ` s ) ) )
87		ovex 6116	. . . . . 6 |- ( s A 0 ) e. _V
88	87	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 0 ) e. _V )
89	86, 88	mprg 2785	. . . 4 |- ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) <-> A. s e. ( 0 [,] 1 ) ( s A 0 ) = ( M ` s ) )
90	85, 89	sylib 196	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( s A 0 ) = ( M ` s ) )
91	90	r19.21bi 2814	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) = ( M ` s ) )
92		1elunit 11404	. . . . . . . . . . . . . 14 |- 1 e. ( 0 [,] 1 )
93		opelxpi 4871	. . . . . . . . . . . . . 14 |- ( ( s e. ( 0 [,] 1 ) /\ 1 e. ( 0 [,] 1 ) ) -> <. s , 1 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
94	92, 93	mpan2 671	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. s , 1 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
95		fvco3 5768	. . . . . . . . . . . . 13 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ <. s , 1 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( F ` ( A ` <. s , 1 >. ) ) )
96	22, 94, 95	syl2an 477	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( F ` ( A ` <. s , 1 >. ) ) )
97	29	fveq1d 5693	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( K ` <. s , 1 >. ) )
98	96, 97	eqtr3d 2477	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 1 >. ) ) = ( K ` <. s , 1 >. ) )
99		df-ov 6094	. . . . . . . . . . . 12 |- ( s A 1 ) = ( A ` <. s , 1 >. )
100	99	fveq2i 5694	. . . . . . . . . . 11 |- ( F ` ( s A 1 ) ) = ( F ` ( A ` <. s , 1 >. ) )
101		df-ov 6094	. . . . . . . . . . 11 |- ( s K 1 ) = ( K ` <. s , 1 >. )
102	98, 100, 101	3eqtr4g 2500	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( s K 1 ) )
103	40	simprd 463	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s K 1 ) = ( H ` s ) )
104	102, 103	eqtrd 2475	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( H ` s ) )
105	104	mpteq2dva 4378	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
106		fovrn 6233	. . . . . . . . . . 11 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) /\ 1 e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
107	92, 106	mp3an3 1303	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
108	22, 107	sylan 471	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
109		eqidd 2444	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
110		fveq2 5691	. . . . . . . . 9 |- ( x = ( s A 1 ) -> ( F ` x ) = ( F ` ( s A 1 ) ) )
111	108, 109, 53, 110	fmptco 5876	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) )
112	19, 50	cnf 18850	. . . . . . . . . 10 |- ( H e. ( II Cn J ) -> H : ( 0 [,] 1 ) --> U. J )
113	10, 112	syl 16	. . . . . . . . 9 |- ( ph -> H : ( 0 [,] 1 ) --> U. J )
114	113	feqmptd 5744	. . . . . . . 8 |- ( ph -> H = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
115	105, 111, 114	3eqtr4d 2485	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H )
116		iicon 20463	. . . . . . . . . . . . 13 |- II e. Con
117	116	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. Con )
118		iinllycon 27143	. . . . . . . . . . . . 13 |- II e. 𝑛Locally Con
119	118	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. 𝑛Locally Con )
120	37, 63, 61, 17	cnmpt12f 19239	. . . . . . . . . . . 12 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) e. ( II Cn C ) )
121		cvmtop1 27149	. . . . . . . . . . . . . . 15 |- ( F e. ( C CovMap J ) -> C e. Top )
122	3, 121	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> C e. Top )
123	1	toptopon 18538	. . . . . . . . . . . . . 14 |- ( C e. Top <-> C e. (TopOn ` B ) )
124	122, 123	sylib 196	. . . . . . . . . . . . 13 |- ( ph -> C e. (TopOn ` B ) )
125		ffvelrn 5841	. . . . . . . . . . . . . 14 |- ( ( M : ( 0 [,] 1 ) --> B /\ 0 e. ( 0 [,] 1 ) ) -> ( M ` 0 ) e. B )
126	83, 23, 125	sylancl 662	. . . . . . . . . . . . 13 |- ( ph -> ( M ` 0 ) e. B )
127		cnconst2 18887	. . . . . . . . . . . . 13 |- ( ( II e. (TopOn ` ( 0 [,] 1 ) ) /\ C e. (TopOn ` B ) /\ ( M ` 0 ) e. B ) -> ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) e. ( II Cn C ) )
128	37, 124, 126, 127	syl3anc 1218	. . . . . . . . . . . 12 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) e. ( II Cn C ) )
129	4, 10, 11	phtpyi 20556	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 K s ) = ( G ` 0 ) /\ ( 1 K s ) = ( G ` 1 ) ) )
130	129	simpld 459	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 K s ) = ( G ` 0 ) )
131		opelxpi 4871	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 0 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
132	23, 131	mpan 670	. . . . . . . . . . . . . . . . . . 19 |- ( s e. ( 0 [,] 1 ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
133		fvco3 5768	. . . . . . . . . . . . . . . . . . 19 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( F ` ( A ` <. 0 , s >. ) ) )
134	22, 132, 133	syl2an 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( F ` ( A ` <. 0 , s >. ) ) )
135	29	fveq1d 5693	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( K ` <. 0 , s >. ) )
136	134, 135	eqtr3d 2477	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 0 , s >. ) ) = ( K ` <. 0 , s >. ) )
137		df-ov 6094	. . . . . . . . . . . . . . . . . 18 |- ( 0 A s ) = ( A ` <. 0 , s >. )
138	137	fveq2i 5694	. . . . . . . . . . . . . . . . 17 |- ( F ` ( 0 A s ) ) = ( F ` ( A ` <. 0 , s >. ) )
139		df-ov 6094	. . . . . . . . . . . . . . . . 17 |- ( 0 K s ) = ( K ` <. 0 , s >. )
140	136, 138, 139	3eqtr4g 2500	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( 0 K s ) )
141	7	simp3d 1002	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( M ` 0 ) = P )
142	141	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( M ` 0 ) = P )
143	142	fveq2d 5695	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( F ` P ) )
144	6	adantr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` P ) = ( G ` 0 ) )
145	143, 144	eqtrd 2475	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( G ` 0 ) )
146	130, 140, 145	3eqtr4d 2485	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( F ` ( M ` 0 ) ) )
147	146	mpteq2dva 4378	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) ) )
148		fconstmpt 4882	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) )
149	147, 148	syl6eqr 2493	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
150		fovrn 6233	. . . . . . . . . . . . . . . 16 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ 0 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
151	23, 150	mp3an2 1302	. . . . . . . . . . . . . . 15 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
152	22, 151	sylan 471	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
153		eqidd 2444	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) )
154		fveq2 5691	. . . . . . . . . . . . . 14 |- ( x = ( 0 A s ) -> ( F ` x ) = ( F ` ( 0 A s ) ) )
155	152, 153, 53, 154	fmptco 5876	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) )
156		ffn 5559	. . . . . . . . . . . . . . 15 |- ( F : B --> U. J -> F Fn B )
157	52, 156	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> F Fn B )
158		fcoconst 5880	. . . . . . . . . . . . . 14 |- ( ( F Fn B /\ ( M ` 0 ) e. B ) -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
159	157, 126, 158	syl2anc 661	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
160	149, 155, 159	3eqtr4d 2485	. . . . . . . . . . . 12 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) )
161	60, 141	eqtr4d 2478	. . . . . . . . . . . . 13 |- ( ph -> ( 0 A 0 ) = ( M ` 0 ) )
162		oveq2 6099	. . . . . . . . . . . . . . 15 |- ( s = 0 -> ( 0 A s ) = ( 0 A 0 ) )
163		eqid 2443	. . . . . . . . . . . . . . 15 |- ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) )
164	162, 163, 72	fvmpt 5774	. . . . . . . . . . . . . 14 |- ( 0 e. ( 0 [,] 1 ) -> ( ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ` 0 ) = ( 0 A 0 ) )
165	23, 164	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ` 0 ) = ( 0 A 0 )
166		fvex 5701	. . . . . . . . . . . . . . 15 |- ( M ` 0 ) e. _V
167	166	fvconst2 5933	. . . . . . . . . . . . . 14 |- ( 0 e. ( 0 [,] 1 ) -> ( ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ` 0 ) = ( M ` 0 ) )
168	23, 167	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ` 0 ) = ( M ` 0 )
169	161, 165, 168	3eqtr4g 2500	. . . . . . . . . . . 12 |- ( ph -> ( ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ` 0 ) = ( ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ` 0 ) )
170	1, 19, 3, 117, 119, 62, 120, 128, 160, 169	cvmliftmoi 27172	. . . . . . . . . . 11 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) )
171		fconstmpt 4882	. . . . . . . . . . 11 |- ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) )
172	170, 171	syl6eq 2491	. . . . . . . . . 10 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) ) )
173		mpteqb 5788	. . . . . . . . . . 11 |- ( A. s e. ( 0 [,] 1 ) ( 0 A s ) e. _V -> ( ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) ) <-> A. s e. ( 0 [,] 1 ) ( 0 A s ) = ( M ` 0 ) ) )
174		ovex 6116	. . . . . . . . . . . 12 |- ( 0 A s ) e. _V
175	174	a1i 11	. . . . . . . . . . 11 |- ( s e. ( 0 [,] 1 ) -> ( 0 A s ) e. _V )
176	173, 175	mprg 2785	. . . . . . . . . 10 |- ( ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) ) <-> A. s e. ( 0 [,] 1 ) ( 0 A s ) = ( M ` 0 ) )
177	172, 176	sylib 196	. . . . . . . . 9 |- ( ph -> A. s e. ( 0 [,] 1 ) ( 0 A s ) = ( M ` 0 ) )
178		oveq2 6099	. . . . . . . . . . 11 |- ( s = 1 -> ( 0 A s ) = ( 0 A 1 ) )
179	178	eqeq1d 2451	. . . . . . . . . 10 |- ( s = 1 -> ( ( 0 A s ) = ( M ` 0 ) <-> ( 0 A 1 ) = ( M ` 0 ) ) )
180	179	rspcv 3069	. . . . . . . . 9 |- ( 1 e. ( 0 [,] 1 ) -> ( A. s e. ( 0 [,] 1 ) ( 0 A s ) = ( M ` 0 ) -> ( 0 A 1 ) = ( M ` 0 ) ) )
181	92, 177, 180	mpsyl 63	. . . . . . . 8 |- ( ph -> ( 0 A 1 ) = ( M ` 0 ) )
182	181, 141	eqtrd 2475	. . . . . . 7 |- ( ph -> ( 0 A 1 ) = P )
183	92	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 e. ( 0 [,] 1 ) )
184	37, 37, 183	cnmptc 19235	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 1 ) e. ( II Cn II ) )
185	37, 61, 184, 17	cnmpt12f 19239	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) e. ( II Cn C ) )
186	1	cvmlift 27188	. . . . . . . . 9 |- ( ( ( F e. ( C CovMap J ) /\ H e. ( II Cn J ) ) /\ ( P e. B /\ ( F ` P ) = ( H ` 0 ) ) ) -> E! f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
187	3, 10, 5, 14, 186	syl22anc 1219	. . . . . . . 8 |- ( ph -> E! f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
188		coeq2 4998	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( F o. f ) = ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) )
189	188	eqeq1d 2451	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( ( F o. f ) = H <-> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H ) )
190		fveq1 5690	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) )
191		oveq1 6098	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 1 ) = ( 0 A 1 ) )
192		eqid 2443	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) )
193		ovex 6116	. . . . . . . . . . . . . 14 |- ( 0 A 1 ) e. _V
194	191, 192, 193	fvmpt 5774	. . . . . . . . . . . . 13 |- ( 0 e. ( 0 [,] 1 ) -> ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) = ( 0 A 1 ) )
195	23, 194	ax-mp 5	. . . . . . . . . . . 12 |- ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) = ( 0 A 1 )
196	190, 195	syl6eq 2491	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( 0 A 1 ) )
197	196	eqeq1d 2451	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( ( f ` 0 ) = P <-> ( 0 A 1 ) = P ) )
198	189, 197	anbi12d 710	. . . . . . . . 9 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( ( ( F o. f ) = H /\ ( f ` 0 ) = P ) <-> ( ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H /\ ( 0 A 1 ) = P ) ) )
199	198	riota2 6075	. . . . . . . 8 |- ( ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) e. ( II Cn C ) /\ E! f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) ) -> ( ( ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H /\ ( 0 A 1 ) = P ) <-> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) )
200	185, 187, 199	syl2anc 661	. . . . . . 7 |- ( ph -> ( ( ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H /\ ( 0 A 1 ) = P ) <-> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) )
201	115, 182, 200	mpbi2and 912	. . . . . 6 |- ( ph -> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
202	9, 201	syl5eq 2487	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
203	19, 1	cnf 18850	. . . . . . 7 |- ( N e. ( II Cn C ) -> N : ( 0 [,] 1 ) --> B )
204	16, 203	syl 16	. . . . . 6 |- ( ph -> N : ( 0 [,] 1 ) --> B )
205	204	feqmptd 5744	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
206	202, 205	eqtr3d 2477	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
207		mpteqb 5788	. . . . 5 |- ( A. s e. ( 0 [,] 1 ) ( s A 1 ) e. _V -> ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) <-> A. s e. ( 0 [,] 1 ) ( s A 1 ) = ( N ` s ) ) )
208		ovex 6116	. . . . . 6 |- ( s A 1 ) e. _V
209	208	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 1 ) e. _V )
210	207, 209	mprg 2785	. . . 4 |- ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) <-> A. s e. ( 0 [,] 1 ) ( s A 1 ) = ( N ` s ) )
211	206, 210	sylib 196	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( s A 1 ) = ( N ` s ) )
212	211	r19.21bi 2814	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) = ( N ` s ) )
213	177	r19.21bi 2814	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) = ( M ` 0 ) )
214	37, 184, 61, 17	cnmpt12f 19239	. . . . . 6 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) e. ( II Cn C ) )
215		ffvelrn 5841	. . . . . . . 8 |- ( ( M : ( 0 [,] 1 ) --> B /\ 1 e. ( 0 [,] 1 ) ) -> ( M ` 1 ) e. B )
216	83, 92, 215	sylancl 662	. . . . . . 7 |- ( ph -> ( M ` 1 ) e. B )
217		cnconst2 18887	. . . . . . 7 |- ( ( II e. (TopOn ` ( 0 [,] 1 ) ) /\ C e. (TopOn ` B ) /\ ( M ` 1 ) e. B ) -> ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) e. ( II Cn C ) )
218	37, 124, 216, 217	syl3anc 1218	. . . . . 6 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) e. ( II Cn C ) )
219		opelxpi 4871	. . . . . . . . . . . . . 14 |- ( ( 1 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
220	92, 219	mpan 670	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
221		fvco3 5768	. . . . . . . . . . . . 13 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( F ` ( A ` <. 1 , s >. ) ) )
222	22, 220, 221	syl2an 477	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( F ` ( A ` <. 1 , s >. ) ) )
223	29	fveq1d 5693	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( K ` <. 1 , s >. ) )
224	222, 223	eqtr3d 2477	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 1 , s >. ) ) = ( K ` <. 1 , s >. ) )
225		df-ov 6094	. . . . . . . . . . . 12 |- ( 1 A s ) = ( A ` <. 1 , s >. )
226	225	fveq2i 5694	. . . . . . . . . . 11 |- ( F ` ( 1 A s ) ) = ( F ` ( A ` <. 1 , s >. ) )
227		df-ov 6094	. . . . . . . . . . 11 |- ( 1 K s ) = ( K ` <. 1 , s >. )
228	224, 226, 227	3eqtr4g 2500	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( 1 K s ) )
229	129	simprd 463	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 K s ) = ( G ` 1 ) )
230	7	simp2d 1001	. . . . . . . . . . . . 13 |- ( ph -> ( F o. M ) = G )
231	230	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. M ) = G )
232	231	fveq1d 5693	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( G ` 1 ) )
233	83	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> M : ( 0 [,] 1 ) --> B )
234		fvco3 5768	. . . . . . . . . . . 12 |- ( ( M : ( 0 [,] 1 ) --> B /\ 1 e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( F ` ( M ` 1 ) ) )
235	233, 92, 234	sylancl 662	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( F ` ( M ` 1 ) ) )
236	232, 235	eqtr3d 2477	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` 1 ) = ( F ` ( M ` 1 ) ) )
237	228, 229, 236	3eqtrd 2479	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( F ` ( M ` 1 ) ) )
238	237	mpteq2dva 4378	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) ) )
239		fconstmpt 4882	. . . . . . . 8 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) )
240	238, 239	syl6eqr 2493	. . . . . . 7 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
241		fovrn 6233	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ 1 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
242	92, 241	mp3an2 1302	. . . . . . . . 9 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
243	22, 242	sylan 471	. . . . . . . 8 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
244		eqidd 2444	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) )
245		fveq2 5691	. . . . . . . 8 |- ( x = ( 1 A s ) -> ( F ` x ) = ( F ` ( 1 A s ) ) )
246	243, 244, 53, 245	fmptco 5876	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) )
247		fcoconst 5880	. . . . . . . 8 |- ( ( F Fn B /\ ( M ` 1 ) e. B ) -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
248	157, 216, 247	syl2anc 661	. . . . . . 7 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
249	240, 246, 248	3eqtr4d 2485	. . . . . 6 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) )
250		oveq1 6098	. . . . . . . . . 10 |- ( s = 1 -> ( s A 0 ) = ( 1 A 0 ) )
251		fveq2 5691	. . . . . . . . . 10 |- ( s = 1 -> ( M ` s ) = ( M ` 1 ) )
252	250, 251	eqeq12d 2457	. . . . . . . . 9 |- ( s = 1 -> ( ( s A 0 ) = ( M ` s ) <-> ( 1 A 0 ) = ( M ` 1 ) ) )
253	252	rspcv 3069	. . . . . . . 8 |- ( 1 e. ( 0 [,] 1 ) -> ( A. s e. ( 0 [,] 1 ) ( s A 0 ) = ( M ` s ) -> ( 1 A 0 ) = ( M ` 1 ) ) )
254	92, 90, 253	mpsyl 63	. . . . . . 7 |- ( ph -> ( 1 A 0 ) = ( M ` 1 ) )
255		oveq2 6099	. . . . . . . . 9 |- ( s = 0 -> ( 1 A s ) = ( 1 A 0 ) )
256		eqid 2443	. . . . . . . . 9 |- ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) )
257		ovex 6116	. . . . . . . . 9 |- ( 1 A 0 ) e. _V
258	255, 256, 257	fvmpt 5774	. . . . . . . 8 |- ( 0 e. ( 0 [,] 1 ) -> ( ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ` 0 ) = ( 1 A 0 ) )
259	23, 258	ax-mp 5	. . . . . . 7 |- ( ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ` 0 ) = ( 1 A 0 )
260		fvex 5701	. . . . . . . . 9 |- ( M ` 1 ) e. _V
261	260	fvconst2 5933	. . . . . . . 8 |- ( 0 e. ( 0 [,] 1 ) -> ( ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ` 0 ) = ( M ` 1 ) )
262	23, 261	ax-mp 5	. . . . . . 7 |- ( ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ` 0 ) = ( M ` 1 )
263	254, 259, 262	3eqtr4g 2500	. . . . . 6 |- ( ph -> ( ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ` 0 ) = ( ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ` 0 ) )
264	1, 19, 3, 117, 119, 62, 214, 218, 249, 263	cvmliftmoi 27172	. . . . 5 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) )
265		fconstmpt 4882	. . . . 5 |- ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) )
266	264, 265	syl6eq 2491	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) ) )
267		mpteqb 5788	. . . . 5 |- ( A. s e. ( 0 [,] 1 ) ( 1 A s ) e. _V -> ( ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) ) <-> A. s e. ( 0 [,] 1 ) ( 1 A s ) = ( M ` 1 ) ) )
268		ovex 6116	. . . . . 6 |- ( 1 A s ) e. _V
269	268	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( 1 A s ) e. _V )
270	267, 269	mprg 2785	. . . 4 |- ( ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) ) <-> A. s e. ( 0 [,] 1 ) ( 1 A s ) = ( M ` 1 ) )
271	266, 270	sylib 196	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( 1 A s ) = ( M ` 1 ) )
272	271	r19.21bi 2814	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) = ( M ` 1 ) )
273	8, 16, 17, 91, 212, 213, 272	isphtpy2d 20559	1 |- ( ph -> A e. ( M ( PHtpy ` C ) N ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   A.wral 2715   E!wreu 2717   _Vcvv 2972   {csn 3877   <.cop 3883   U.cuni 4091   e. cmpt 4350   X. cxp 4838   o. ccom 4844   Fn wfn 5413   -->wf 5414   ` cfv 5418   iota_crio 6051  (class class class)co 6091   0cc0 9282   1c1 9283   [,]cicc 11303   Topctop 18498  TopOnctopon 18499   Cn ccn 18828   Conccon 19015  𝑛Locally cnlly 19069   tX ctx 19133   IIcii 20451   Htpy chtpy 20539   PHtpycphtpy 20540   CovMap ccvm 27144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-cn 18831  df-cnp 18832  df-cmp 18990  df-con 19016  df-lly 19070  df-nlly 19071  df-tx 19135  df-hmeo 19328  df-xms 19895  df-ms 19896  df-tms 19897  df-ii 20453  df-htpy 20542  df-phtpy 20543  df-phtpc 20564  df-pcon 27110  df-scon 27111  df-cvm 27145

This theorem is referenced by:  cvmliftpht  27207