Theorem cvmliftphtlem 24957

Description: Lemma for cvmliftpht 24958. (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 24941	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
8	7	simp1d 969	. 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 18963	. . . . . 6 |- ( ph -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
13	12	simpld 446	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
14	6, 13	eqtrd 2436	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
15	1, 9, 3, 10, 5, 14	cvmliftiota 24941	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
16	15	simp1d 969	. 2 |- ( ph -> N e. ( II Cn C ) )
17		cvmliftphtlem.a	. 2 |- ( ph -> A e. ( ( II tX II ) Cn C ) )
18		iitop 18863	. . . . . . . . . . . . . . . 16 |- II e. Top
19		iiuni 18864	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) = U. II
20	18, 18, 19, 19	txunii 17578	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
21	20, 1	cnf 17264	. . . . . . . . . . . . . 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 10971	. . . . . . . . . . . . . 14 |- 0 e. ( 0 [,] 1 )
24		opelxpi 4869	. . . . . . . . . . . . . 14 |- ( ( s e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> <. s , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
25	23, 24	mpan2 653	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. s , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
26		fvco3 5759	. . . . . . . . . . . . 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 464	. . . . . . . . . . . 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 452	. . . . . . . . . . . . 13 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. A ) = K )
30	29	fveq1d 5689	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 0 >. ) = ( K ` <. s , 0 >. ) )
31	27, 30	eqtr3d 2438	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 0 >. ) ) = ( K ` <. s , 0 >. ) )
32		df-ov 6043	. . . . . . . . . . . 12 |- ( s A 0 ) = ( A ` <. s , 0 >. )
33	32	fveq2i 5690	. . . . . . . . . . 11 |- ( F ` ( s A 0 ) ) = ( F ` ( A ` <. s , 0 >. ) )
34		df-ov 6043	. . . . . . . . . . 11 |- ( s K 0 ) = ( K ` <. s , 0 >. )
35	31, 33, 34	3eqtr4g 2461	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( s K 0 ) )
36		iitopon 18862	. . . . . . . . . . . . 13 |- II e. (TopOn ` ( 0 [,] 1 ) )
37	36	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
38	4, 10	phtpyhtpy 18960	. . . . . . . . . . . . 13 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( G ( II Htpy J ) H ) )
39	38, 11	sseldd 3309	. . . . . . . . . . . 12 |- ( ph -> K e. ( G ( II Htpy J ) H ) )
40	37, 4, 10, 39	htpyi 18952	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( s K 0 ) = ( G ` s ) /\ ( s K 1 ) = ( H ` s ) ) )
41	40	simpld 446	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s K 0 ) = ( G ` s ) )
42	35, 41	eqtrd 2436	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( G ` s ) )
43	42	mpteq2dva 4255	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
44		fovrn 6175	. . . . . . . . . . 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 1268	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
46	22, 45	sylan 458	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
47		eqidd 2405	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
48		cvmcn 24902	. . . . . . . . . . . 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 2404	. . . . . . . . . . . 12 |- U. J = U. J
51	1, 50	cnf 17264	. . . . . . . . . . 11 |- ( F e. ( C Cn J ) -> F : B --> U. J )
52	49, 51	syl 16	. . . . . . . . . 10 |- ( ph -> F : B --> U. J )
53	52	feqmptd 5738	. . . . . . . . 9 |- ( ph -> F = ( x e. B |-> ( F ` x ) ) )
54		fveq2 5687	. . . . . . . . 9 |- ( x = ( s A 0 ) -> ( F ` x ) = ( F ` ( s A 0 ) ) )
55	46, 47, 53, 54	fmptco 5860	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) )
56	19, 50	cnf 17264	. . . . . . . . . 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 5738	. . . . . . . 8 |- ( ph -> G = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
59	43, 55, 58	3eqtr4d 2446	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G )
60		cvmliftphtlem.0	. . . . . . 7 |- ( ph -> ( 0 A 0 ) = P )
61	37	cnmptid 17646	. . . . . . . . 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 17647	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
64	37, 61, 63, 17	cnmpt12f 17651	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) e. ( II Cn C ) )
65	1	cvmlift 24939	. . . . . . . . 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 1185	. . . . . . . 8 |- ( ph -> E! f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
67		coeq2 4990	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( F o. f ) = ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) )
68	67	eqeq1d 2412	. . . . . . . . . 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 5686	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) )
70		oveq1 6047	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 0 ) = ( 0 A 0 ) )
71		eqid 2404	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) )
72		ovex 6065	. . . . . . . . . . . . . 14 |- ( 0 A 0 ) e. _V
73	70, 71, 72	fvmpt 5765	. . . . . . . . . . . . 13 |- ( 0 e. ( 0 [,] 1 ) -> ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) = ( 0 A 0 ) )
74	23, 73	ax-mp 8	. . . . . . . . . . . 12 |- ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) = ( 0 A 0 )
75	69, 74	syl6eq 2452	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( 0 A 0 ) )
76	75	eqeq1d 2412	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( ( f ` 0 ) = P <-> ( 0 A 0 ) = P ) )
77	68, 76	anbi12d 692	. . . . . . . . 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 6531	. . . . . . . 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 643	. . . . . . 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 888	. . . . . 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 2448	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
82	19, 1	cnf 17264	. . . . . . 7 |- ( M e. ( II Cn C ) -> M : ( 0 [,] 1 ) --> B )
83	8, 82	syl 16	. . . . . 6 |- ( ph -> M : ( 0 [,] 1 ) --> B )
84	83	feqmptd 5738	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
85	81, 84	eqtr3d 2438	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
86		mpteqb 5778	. . . . 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 6065	. . . . . 6 |- ( s A 0 ) e. _V
88	87	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 0 ) e. _V )
89	86, 88	mprg 2735	. . . 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 189	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( s A 0 ) = ( M ` s ) )
91	90	r19.21bi 2764	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) = ( M ` s ) )
92		1elunit 10972	. . . . . . . . . . . . . 14 |- 1 e. ( 0 [,] 1 )
93		opelxpi 4869	. . . . . . . . . . . . . 14 |- ( ( s e. ( 0 [,] 1 ) /\ 1 e. ( 0 [,] 1 ) ) -> <. s , 1 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
94	92, 93	mpan2 653	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. s , 1 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
95		fvco3 5759	. . . . . . . . . . . . 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 464	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( F ` ( A ` <. s , 1 >. ) ) )
97	29	fveq1d 5689	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( K ` <. s , 1 >. ) )
98	96, 97	eqtr3d 2438	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 1 >. ) ) = ( K ` <. s , 1 >. ) )
99		df-ov 6043	. . . . . . . . . . . 12 |- ( s A 1 ) = ( A ` <. s , 1 >. )
100	99	fveq2i 5690	. . . . . . . . . . 11 |- ( F ` ( s A 1 ) ) = ( F ` ( A ` <. s , 1 >. ) )
101		df-ov 6043	. . . . . . . . . . 11 |- ( s K 1 ) = ( K ` <. s , 1 >. )
102	98, 100, 101	3eqtr4g 2461	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( s K 1 ) )
103	40	simprd 450	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s K 1 ) = ( H ` s ) )
104	102, 103	eqtrd 2436	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( H ` s ) )
105	104	mpteq2dva 4255	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
106		fovrn 6175	. . . . . . . . . . 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 1268	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
108	22, 107	sylan 458	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
109		eqidd 2405	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
110		fveq2 5687	. . . . . . . . 9 |- ( x = ( s A 1 ) -> ( F ` x ) = ( F ` ( s A 1 ) ) )
111	108, 109, 53, 110	fmptco 5860	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) )
112	19, 50	cnf 17264	. . . . . . . . . 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 5738	. . . . . . . 8 |- ( ph -> H = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
115	105, 111, 114	3eqtr4d 2446	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H )
116		iicon 18870	. . . . . . . . . . . . 13 |- II e. Con
117	116	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. Con )
118		iinllycon 24894	. . . . . . . . . . . . 13 |- II e. 𝑛Locally Con
119	118	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. 𝑛Locally Con )
120	37, 63, 61, 17	cnmpt12f 17651	. . . . . . . . . . . 12 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) e. ( II Cn C ) )
121		cvmtop1 24900	. . . . . . . . . . . . . . 15 |- ( F e. ( C CovMap J ) -> C e. Top )
122	3, 121	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> C e. Top )
123	1	toptopon 16953	. . . . . . . . . . . . . 14 |- ( C e. Top <-> C e. (TopOn ` B ) )
124	122, 123	sylib 189	. . . . . . . . . . . . 13 |- ( ph -> C e. (TopOn ` B ) )
125		ffvelrn 5827	. . . . . . . . . . . . . 14 |- ( ( M : ( 0 [,] 1 ) --> B /\ 0 e. ( 0 [,] 1 ) ) -> ( M ` 0 ) e. B )
126	83, 23, 125	sylancl 644	. . . . . . . . . . . . 13 |- ( ph -> ( M ` 0 ) e. B )
127		cnconst2 17301	. . . . . . . . . . . . 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 1184	. . . . . . . . . . . 12 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) e. ( II Cn C ) )
129	4, 10, 11	phtpyi 18962	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 K s ) = ( G ` 0 ) /\ ( 1 K s ) = ( G ` 1 ) ) )
130	129	simpld 446	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 K s ) = ( G ` 0 ) )
131		opelxpi 4869	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 0 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
132	23, 131	mpan 652	. . . . . . . . . . . . . . . . . . 19 |- ( s e. ( 0 [,] 1 ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
133		fvco3 5759	. . . . . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( F ` ( A ` <. 0 , s >. ) ) )
135	29	fveq1d 5689	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( K ` <. 0 , s >. ) )
136	134, 135	eqtr3d 2438	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 0 , s >. ) ) = ( K ` <. 0 , s >. ) )
137		df-ov 6043	. . . . . . . . . . . . . . . . . 18 |- ( 0 A s ) = ( A ` <. 0 , s >. )
138	137	fveq2i 5690	. . . . . . . . . . . . . . . . 17 |- ( F ` ( 0 A s ) ) = ( F ` ( A ` <. 0 , s >. ) )
139		df-ov 6043	. . . . . . . . . . . . . . . . 17 |- ( 0 K s ) = ( K ` <. 0 , s >. )
140	136, 138, 139	3eqtr4g 2461	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( 0 K s ) )
141	7	simp3d 971	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( M ` 0 ) = P )
142	141	adantr 452	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( M ` 0 ) = P )
143	142	fveq2d 5691	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( F ` P ) )
144	6	adantr 452	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` P ) = ( G ` 0 ) )
145	143, 144	eqtrd 2436	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( G ` 0 ) )
146	130, 140, 145	3eqtr4d 2446	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( F ` ( M ` 0 ) ) )
147	146	mpteq2dva 4255	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) ) )
148		fconstmpt 4880	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) )
149	147, 148	syl6eqr 2454	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
150		fovrn 6175	. . . . . . . . . . . . . . . 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 1267	. . . . . . . . . . . . . . 15 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
152	22, 151	sylan 458	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
153		eqidd 2405	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) )
154		fveq2 5687	. . . . . . . . . . . . . 14 |- ( x = ( 0 A s ) -> ( F ` x ) = ( F ` ( 0 A s ) ) )
155	152, 153, 53, 154	fmptco 5860	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) )
156		ffn 5550	. . . . . . . . . . . . . . 15 |- ( F : B --> U. J -> F Fn B )
157	52, 156	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> F Fn B )
158		fcoconst 5864	. . . . . . . . . . . . . 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 643	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
160	149, 155, 159	3eqtr4d 2446	. . . . . . . . . . . 12 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) )
161	60, 141	eqtr4d 2439	. . . . . . . . . . . . 13 |- ( ph -> ( 0 A 0 ) = ( M ` 0 ) )
162		oveq2 6048	. . . . . . . . . . . . . . 15 |- ( s = 0 -> ( 0 A s ) = ( 0 A 0 ) )
163		eqid 2404	. . . . . . . . . . . . . . 15 |- ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) )
164	162, 163, 72	fvmpt 5765	. . . . . . . . . . . . . 14 |- ( 0 e. ( 0 [,] 1 ) -> ( ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ` 0 ) = ( 0 A 0 ) )
165	23, 164	ax-mp 8	. . . . . . . . . . . . 13 |- ( ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ` 0 ) = ( 0 A 0 )
166		fvex 5701	. . . . . . . . . . . . . . 15 |- ( M ` 0 ) e. _V
167	166	fvconst2 5906	. . . . . . . . . . . . . 14 |- ( 0 e. ( 0 [,] 1 ) -> ( ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ` 0 ) = ( M ` 0 ) )
168	23, 167	ax-mp 8	. . . . . . . . . . . . 13 |- ( ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ` 0 ) = ( M ` 0 )
169	161, 165, 168	3eqtr4g 2461	. . . . . . . . . . . 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 24923	. . . . . . . . . . 11 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) )
171		fconstmpt 4880	. . . . . . . . . . 11 |- ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) )
172	170, 171	syl6eq 2452	. . . . . . . . . 10 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) ) )
173		mpteqb 5778	. . . . . . . . . . 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 6065	. . . . . . . . . . . 12 |- ( 0 A s ) e. _V
175	174	a1i 11	. . . . . . . . . . 11 |- ( s e. ( 0 [,] 1 ) -> ( 0 A s ) e. _V )
176	173, 175	mprg 2735	. . . . . . . . . 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 189	. . . . . . . . 9 |- ( ph -> A. s e. ( 0 [,] 1 ) ( 0 A s ) = ( M ` 0 ) )
178		oveq2 6048	. . . . . . . . . . 11 |- ( s = 1 -> ( 0 A s ) = ( 0 A 1 ) )
179	178	eqeq1d 2412	. . . . . . . . . 10 |- ( s = 1 -> ( ( 0 A s ) = ( M ` 0 ) <-> ( 0 A 1 ) = ( M ` 0 ) ) )
180	179	rspcv 3008	. . . . . . . . 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 61	. . . . . . . 8 |- ( ph -> ( 0 A 1 ) = ( M ` 0 ) )
182	181, 141	eqtrd 2436	. . . . . . 7 |- ( ph -> ( 0 A 1 ) = P )
183	92	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 e. ( 0 [,] 1 ) )
184	37, 37, 183	cnmptc 17647	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 1 ) e. ( II Cn II ) )
185	37, 61, 184, 17	cnmpt12f 17651	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) e. ( II Cn C ) )
186	1	cvmlift 24939	. . . . . . . . 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 1185	. . . . . . . 8 |- ( ph -> E! f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
188		coeq2 4990	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( F o. f ) = ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) )
189	188	eqeq1d 2412	. . . . . . . . . 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 5686	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) )
191		oveq1 6047	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 1 ) = ( 0 A 1 ) )
192		eqid 2404	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) )
193		ovex 6065	. . . . . . . . . . . . . 14 |- ( 0 A 1 ) e. _V
194	191, 192, 193	fvmpt 5765	. . . . . . . . . . . . 13 |- ( 0 e. ( 0 [,] 1 ) -> ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) = ( 0 A 1 ) )
195	23, 194	ax-mp 8	. . . . . . . . . . . 12 |- ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) = ( 0 A 1 )
196	190, 195	syl6eq 2452	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( 0 A 1 ) )
197	196	eqeq1d 2412	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( ( f ` 0 ) = P <-> ( 0 A 1 ) = P ) )
198	189, 197	anbi12d 692	. . . . . . . . 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 6531	. . . . . . . 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 643	. . . . . . 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 888	. . . . . 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 2448	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
203	19, 1	cnf 17264	. . . . . . 7 |- ( N e. ( II Cn C ) -> N : ( 0 [,] 1 ) --> B )
204	16, 203	syl 16	. . . . . 6 |- ( ph -> N : ( 0 [,] 1 ) --> B )
205	204	feqmptd 5738	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
206	202, 205	eqtr3d 2438	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
207		mpteqb 5778	. . . . 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 6065	. . . . . 6 |- ( s A 1 ) e. _V
209	208	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 1 ) e. _V )
210	207, 209	mprg 2735	. . . 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 189	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( s A 1 ) = ( N ` s ) )
212	211	r19.21bi 2764	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) = ( N ` s ) )
213	177	r19.21bi 2764	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) = ( M ` 0 ) )
214	37, 184, 61, 17	cnmpt12f 17651	. . . . . 6 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) e. ( II Cn C ) )
215		ffvelrn 5827	. . . . . . . 8 |- ( ( M : ( 0 [,] 1 ) --> B /\ 1 e. ( 0 [,] 1 ) ) -> ( M ` 1 ) e. B )
216	83, 92, 215	sylancl 644	. . . . . . 7 |- ( ph -> ( M ` 1 ) e. B )
217		cnconst2 17301	. . . . . . 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 1184	. . . . . 6 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) e. ( II Cn C ) )
219		opelxpi 4869	. . . . . . . . . . . . . 14 |- ( ( 1 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
220	92, 219	mpan 652	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
221		fvco3 5759	. . . . . . . . . . . . 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 464	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( F ` ( A ` <. 1 , s >. ) ) )
223	29	fveq1d 5689	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( K ` <. 1 , s >. ) )
224	222, 223	eqtr3d 2438	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 1 , s >. ) ) = ( K ` <. 1 , s >. ) )
225		df-ov 6043	. . . . . . . . . . . 12 |- ( 1 A s ) = ( A ` <. 1 , s >. )
226	225	fveq2i 5690	. . . . . . . . . . 11 |- ( F ` ( 1 A s ) ) = ( F ` ( A ` <. 1 , s >. ) )
227		df-ov 6043	. . . . . . . . . . 11 |- ( 1 K s ) = ( K ` <. 1 , s >. )
228	224, 226, 227	3eqtr4g 2461	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( 1 K s ) )
229	129	simprd 450	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 K s ) = ( G ` 1 ) )
230	7	simp2d 970	. . . . . . . . . . . . 13 |- ( ph -> ( F o. M ) = G )
231	230	adantr 452	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. M ) = G )
232	231	fveq1d 5689	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( G ` 1 ) )
233	83	adantr 452	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> M : ( 0 [,] 1 ) --> B )
234		fvco3 5759	. . . . . . . . . . . 12 |- ( ( M : ( 0 [,] 1 ) --> B /\ 1 e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( F ` ( M ` 1 ) ) )
235	233, 92, 234	sylancl 644	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( F ` ( M ` 1 ) ) )
236	232, 235	eqtr3d 2438	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` 1 ) = ( F ` ( M ` 1 ) ) )
237	228, 229, 236	3eqtrd 2440	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( F ` ( M ` 1 ) ) )
238	237	mpteq2dva 4255	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) ) )
239		fconstmpt 4880	. . . . . . . 8 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) )
240	238, 239	syl6eqr 2454	. . . . . . 7 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
241		fovrn 6175	. . . . . . . . . 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 1267	. . . . . . . . 9 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
243	22, 242	sylan 458	. . . . . . . 8 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
244		eqidd 2405	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) )
245		fveq2 5687	. . . . . . . 8 |- ( x = ( 1 A s ) -> ( F ` x ) = ( F ` ( 1 A s ) ) )
246	243, 244, 53, 245	fmptco 5860	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) )
247		fcoconst 5864	. . . . . . . 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 643	. . . . . . 7 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
249	240, 246, 248	3eqtr4d 2446	. . . . . 6 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) )
250		oveq1 6047	. . . . . . . . . 10 |- ( s = 1 -> ( s A 0 ) = ( 1 A 0 ) )
251		fveq2 5687	. . . . . . . . . 10 |- ( s = 1 -> ( M ` s ) = ( M ` 1 ) )
252	250, 251	eqeq12d 2418	. . . . . . . . 9 |- ( s = 1 -> ( ( s A 0 ) = ( M ` s ) <-> ( 1 A 0 ) = ( M ` 1 ) ) )
253	252	rspcv 3008	. . . . . . . 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 61	. . . . . . 7 |- ( ph -> ( 1 A 0 ) = ( M ` 1 ) )
255		oveq2 6048	. . . . . . . . 9 |- ( s = 0 -> ( 1 A s ) = ( 1 A 0 ) )
256		eqid 2404	. . . . . . . . 9 |- ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) )
257		ovex 6065	. . . . . . . . 9 |- ( 1 A 0 ) e. _V
258	255, 256, 257	fvmpt 5765	. . . . . . . 8 |- ( 0 e. ( 0 [,] 1 ) -> ( ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ` 0 ) = ( 1 A 0 ) )
259	23, 258	ax-mp 8	. . . . . . 7 |- ( ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ` 0 ) = ( 1 A 0 )
260		fvex 5701	. . . . . . . . 9 |- ( M ` 1 ) e. _V
261	260	fvconst2 5906	. . . . . . . 8 |- ( 0 e. ( 0 [,] 1 ) -> ( ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ` 0 ) = ( M ` 1 ) )
262	23, 261	ax-mp 8	. . . . . . 7 |- ( ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ` 0 ) = ( M ` 1 )
263	254, 259, 262	3eqtr4g 2461	. . . . . 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 24923	. . . . 5 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) )
265		fconstmpt 4880	. . . . 5 |- ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) )
266	264, 265	syl6eq 2452	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) ) )
267		mpteqb 5778	. . . . 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 6065	. . . . . 6 |- ( 1 A s ) e. _V
269	268	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( 1 A s ) e. _V )
270	267, 269	mprg 2735	. . . 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 189	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( 1 A s ) = ( M ` 1 ) )
272	271	r19.21bi 2764	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) = ( M ` 1 ) )
273	8, 16, 17, 91, 212, 213, 272	isphtpy2d 18965	1 |- ( ph -> A e. ( M ( PHtpy ` C ) N ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   A.wral 2666   E!wreu 2668   _Vcvv 2916   {csn 3774   <.cop 3777   U.cuni 3975   e. cmpt 4226   X. cxp 4835   o. ccom 4841   Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   iota_crio 6501   0cc0 8946   1c1 8947   [,]cicc 10875   Topctop 16913  TopOnctopon 16914   Cn ccn 17242   Conccon 17427  𝑛Locally cnlly 17481   tX ctx 17545   IIcii 18858   Htpy chtpy 18945   PHtpycphtpy 18946   CovMap ccvm 24895

This theorem is referenced by:  cvmliftpht  24958

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-cnp 17246  df-cmp 17404  df-con 17428  df-lly 17482  df-nlly 17483  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860  df-htpy 18948  df-phtpy 18949  df-phtpc 18970  df-pcon 24861  df-scon 24862  df-cvm 24896