Theorem cvmliftphtlem 30112

Description: Lemma for cvmliftpht 30113. (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 30096	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
8	7	simp1d 1042	. 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 22094	. . . . . 6 |- ( ph -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
13	12	simpld 466	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
14	6, 13	eqtrd 2505	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
15	1, 9, 3, 10, 5, 14	cvmliftiota 30096	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
16	15	simp1d 1042	. 2 |- ( ph -> N e. ( II Cn C ) )
17		cvmliftphtlem.a	. 2 |- ( ph -> A e. ( ( II tX II ) Cn C ) )
18		iitop 21990	. . . . . . . . . . . . . . . 16 |- II e. Top
19		iiuni 21991	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) = U. II
20	18, 18, 19, 19	txunii 20685	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
21	20, 1	cnf 20339	. . . . . . . . . . . . . 14 |- ( A e. ( ( II tX II ) Cn C ) -> A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B )
22	17, 21	syl 17	. . . . . . . . . . . . 13 |- ( ph -> A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B )
23		0elunit 11776	. . . . . . . . . . . . . 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 685	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. s , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
26		fvco3 5957	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 472	. . . . . . . . . . . . 13 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. A ) = K )
30	29	fveq1d 5881	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 0 >. ) = ( K ` <. s , 0 >. ) )
31	27, 30	eqtr3d 2507	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 0 >. ) ) = ( K ` <. s , 0 >. ) )
32		df-ov 6311	. . . . . . . . . . . 12 |- ( s A 0 ) = ( A ` <. s , 0 >. )
33	32	fveq2i 5882	. . . . . . . . . . 11 |- ( F ` ( s A 0 ) ) = ( F ` ( A ` <. s , 0 >. ) )
34		df-ov 6311	. . . . . . . . . . 11 |- ( s K 0 ) = ( K ` <. s , 0 >. )
35	31, 33, 34	3eqtr4g 2530	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( s K 0 ) )
36		iitopon 21989	. . . . . . . . . . . . 13 |- II e. (TopOn ` ( 0 [,] 1 ) )
37	36	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
38	4, 10	phtpyhtpy 22091	. . . . . . . . . . . . 13 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( G ( II Htpy J ) H ) )
39	38, 11	sseldd 3419	. . . . . . . . . . . 12 |- ( ph -> K e. ( G ( II Htpy J ) H ) )
40	37, 4, 10, 39	htpyi 22083	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( s K 0 ) = ( G ` s ) /\ ( s K 1 ) = ( H ` s ) ) )
41	40	simpld 466	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s K 0 ) = ( G ` s ) )
42	35, 41	eqtrd 2505	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( G ` s ) )
43	42	mpteq2dva 4482	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
44		fovrn 6458	. . . . . . . . . . 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 1379	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
46	22, 45	sylan 479	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
47		eqidd 2472	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
48		cvmcn 30057	. . . . . . . . . . . 12 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
49	3, 48	syl 17	. . . . . . . . . . 11 |- ( ph -> F e. ( C Cn J ) )
50		eqid 2471	. . . . . . . . . . . 12 |- U. J = U. J
51	1, 50	cnf 20339	. . . . . . . . . . 11 |- ( F e. ( C Cn J ) -> F : B --> U. J )
52	49, 51	syl 17	. . . . . . . . . 10 |- ( ph -> F : B --> U. J )
53	52	feqmptd 5932	. . . . . . . . 9 |- ( ph -> F = ( x e. B |-> ( F ` x ) ) )
54		fveq2 5879	. . . . . . . . 9 |- ( x = ( s A 0 ) -> ( F ` x ) = ( F ` ( s A 0 ) ) )
55	46, 47, 53, 54	fmptco 6072	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) )
56	19, 50	cnf 20339	. . . . . . . . . 10 |- ( G e. ( II Cn J ) -> G : ( 0 [,] 1 ) --> U. J )
57	4, 56	syl 17	. . . . . . . . 9 |- ( ph -> G : ( 0 [,] 1 ) --> U. J )
58	57	feqmptd 5932	. . . . . . . 8 |- ( ph -> G = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
59	43, 55, 58	3eqtr4d 2515	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G )
60		cvmliftphtlem.0	. . . . . . 7 |- ( ph -> ( 0 A 0 ) = P )
61	37	cnmptid 20753	. . . . . . . . 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 20754	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
64	37, 61, 63, 17	cnmpt12f 20758	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) e. ( II Cn C ) )
65	1	cvmlift 30094	. . . . . . . . 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 1293	. . . . . . . 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 2473	. . . . . . . . . 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 5878	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) )
70		oveq1 6315	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 0 ) = ( 0 A 0 ) )
71		eqid 2471	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) )
72		ovex 6336	. . . . . . . . . . . . . 14 |- ( 0 A 0 ) e. _V
73	70, 71, 72	fvmpt 5963	. . . . . . . . . . . . 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 2521	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( 0 A 0 ) )
76	75	eqeq1d 2473	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( ( f ` 0 ) = P <-> ( 0 A 0 ) = P ) )
77	68, 76	anbi12d 725	. . . . . . . . 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 6292	. . . . . . . 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 673	. . . . . . 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 935	. . . . . 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 2517	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
82	19, 1	cnf 20339	. . . . . . 7 |- ( M e. ( II Cn C ) -> M : ( 0 [,] 1 ) --> B )
83	8, 82	syl 17	. . . . . 6 |- ( ph -> M : ( 0 [,] 1 ) --> B )
84	83	feqmptd 5932	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
85	81, 84	eqtr3d 2507	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
86		mpteqb 5979	. . . . 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 6336	. . . . . 6 |- ( s A 0 ) e. _V
88	87	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 0 ) e. _V )
89	86, 88	mprg 2770	. . . 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 201	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( s A 0 ) = ( M ` s ) )
91	90	r19.21bi 2776	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) = ( M ` s ) )
92		1elunit 11777	. . . . . . . . . . . . . 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 685	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. s , 1 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
95		fvco3 5957	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( F ` ( A ` <. s , 1 >. ) ) )
97	29	fveq1d 5881	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( K ` <. s , 1 >. ) )
98	96, 97	eqtr3d 2507	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 1 >. ) ) = ( K ` <. s , 1 >. ) )
99		df-ov 6311	. . . . . . . . . . . 12 |- ( s A 1 ) = ( A ` <. s , 1 >. )
100	99	fveq2i 5882	. . . . . . . . . . 11 |- ( F ` ( s A 1 ) ) = ( F ` ( A ` <. s , 1 >. ) )
101		df-ov 6311	. . . . . . . . . . 11 |- ( s K 1 ) = ( K ` <. s , 1 >. )
102	98, 100, 101	3eqtr4g 2530	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( s K 1 ) )
103	40	simprd 470	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s K 1 ) = ( H ` s ) )
104	102, 103	eqtrd 2505	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( H ` s ) )
105	104	mpteq2dva 4482	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
106		fovrn 6458	. . . . . . . . . . 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 1379	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
108	22, 107	sylan 479	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
109		eqidd 2472	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
110		fveq2 5879	. . . . . . . . 9 |- ( x = ( s A 1 ) -> ( F ` x ) = ( F ` ( s A 1 ) ) )
111	108, 109, 53, 110	fmptco 6072	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) )
112	19, 50	cnf 20339	. . . . . . . . . 10 |- ( H e. ( II Cn J ) -> H : ( 0 [,] 1 ) --> U. J )
113	10, 112	syl 17	. . . . . . . . 9 |- ( ph -> H : ( 0 [,] 1 ) --> U. J )
114	113	feqmptd 5932	. . . . . . . 8 |- ( ph -> H = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
115	105, 111, 114	3eqtr4d 2515	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H )
116		iicon 21997	. . . . . . . . . . . . 13 |- II e. Con
117	116	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. Con )
118		iinllycon 30049	. . . . . . . . . . . . 13 |- II e. 𝑛Locally Con
119	118	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. 𝑛Locally Con )
120	37, 63, 61, 17	cnmpt12f 20758	. . . . . . . . . . . 12 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) e. ( II Cn C ) )
121		cvmtop1 30055	. . . . . . . . . . . . . . 15 |- ( F e. ( C CovMap J ) -> C e. Top )
122	3, 121	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> C e. Top )
123	1	toptopon 20025	. . . . . . . . . . . . . 14 |- ( C e. Top <-> C e. (TopOn ` B ) )
124	122, 123	sylib 201	. . . . . . . . . . . . 13 |- ( ph -> C e. (TopOn ` B ) )
125		ffvelrn 6035	. . . . . . . . . . . . . 14 |- ( ( M : ( 0 [,] 1 ) --> B /\ 0 e. ( 0 [,] 1 ) ) -> ( M ` 0 ) e. B )
126	83, 23, 125	sylancl 675	. . . . . . . . . . . . 13 |- ( ph -> ( M ` 0 ) e. B )
127		cnconst2 20376	. . . . . . . . . . . . 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 1292	. . . . . . . . . . . 12 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) e. ( II Cn C ) )
129	4, 10, 11	phtpyi 22093	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 K s ) = ( G ` 0 ) /\ ( 1 K s ) = ( G ` 1 ) ) )
130	129	simpld 466	. . . . . . . . . . . . . . . 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 684	. . . . . . . . . . . . . . . . . . 19 |- ( s e. ( 0 [,] 1 ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
133		fvco3 5957	. . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( F ` ( A ` <. 0 , s >. ) ) )
135	29	fveq1d 5881	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( K ` <. 0 , s >. ) )
136	134, 135	eqtr3d 2507	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 0 , s >. ) ) = ( K ` <. 0 , s >. ) )
137		df-ov 6311	. . . . . . . . . . . . . . . . . 18 |- ( 0 A s ) = ( A ` <. 0 , s >. )
138	137	fveq2i 5882	. . . . . . . . . . . . . . . . 17 |- ( F ` ( 0 A s ) ) = ( F ` ( A ` <. 0 , s >. ) )
139		df-ov 6311	. . . . . . . . . . . . . . . . 17 |- ( 0 K s ) = ( K ` <. 0 , s >. )
140	136, 138, 139	3eqtr4g 2530	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( 0 K s ) )
141	7	simp3d 1044	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( M ` 0 ) = P )
142	141	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( M ` 0 ) = P )
143	142	fveq2d 5883	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( F ` P ) )
144	6	adantr 472	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` P ) = ( G ` 0 ) )
145	143, 144	eqtrd 2505	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( G ` 0 ) )
146	130, 140, 145	3eqtr4d 2515	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( F ` ( M ` 0 ) ) )
147	146	mpteq2dva 4482	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) ) )
148		fconstmpt 4883	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) )
149	147, 148	syl6eqr 2523	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
150		fovrn 6458	. . . . . . . . . . . . . . . 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 1378	. . . . . . . . . . . . . . 15 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
152	22, 151	sylan 479	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
153		eqidd 2472	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) )
154		fveq2 5879	. . . . . . . . . . . . . 14 |- ( x = ( 0 A s ) -> ( F ` x ) = ( F ` ( 0 A s ) ) )
155	152, 153, 53, 154	fmptco 6072	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) )
156		ffn 5739	. . . . . . . . . . . . . . 15 |- ( F : B --> U. J -> F Fn B )
157	52, 156	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> F Fn B )
158		fcoconst 6076	. . . . . . . . . . . . . 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 673	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
160	149, 155, 159	3eqtr4d 2515	. . . . . . . . . . . 12 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) )
161	60, 141	eqtr4d 2508	. . . . . . . . . . . . 13 |- ( ph -> ( 0 A 0 ) = ( M ` 0 ) )
162		oveq2 6316	. . . . . . . . . . . . . . 15 |- ( s = 0 -> ( 0 A s ) = ( 0 A 0 ) )
163		eqid 2471	. . . . . . . . . . . . . . 15 |- ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) )
164	162, 163, 72	fvmpt 5963	. . . . . . . . . . . . . 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 5889	. . . . . . . . . . . . . . 15 |- ( M ` 0 ) e. _V
167	166	fvconst2 6136	. . . . . . . . . . . . . 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 2530	. . . . . . . . . . . 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 30078	. . . . . . . . . . 11 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) )
171		fconstmpt 4883	. . . . . . . . . . 11 |- ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) )
172	170, 171	syl6eq 2521	. . . . . . . . . 10 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) ) )
173		mpteqb 5979	. . . . . . . . . . 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 6336	. . . . . . . . . . . 12 |- ( 0 A s ) e. _V
175	174	a1i 11	. . . . . . . . . . 11 |- ( s e. ( 0 [,] 1 ) -> ( 0 A s ) e. _V )
176	173, 175	mprg 2770	. . . . . . . . . 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 201	. . . . . . . . 9 |- ( ph -> A. s e. ( 0 [,] 1 ) ( 0 A s ) = ( M ` 0 ) )
178		oveq2 6316	. . . . . . . . . . 11 |- ( s = 1 -> ( 0 A s ) = ( 0 A 1 ) )
179	178	eqeq1d 2473	. . . . . . . . . 10 |- ( s = 1 -> ( ( 0 A s ) = ( M ` 0 ) <-> ( 0 A 1 ) = ( M ` 0 ) ) )
180	179	rspcv 3132	. . . . . . . . 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 64	. . . . . . . 8 |- ( ph -> ( 0 A 1 ) = ( M ` 0 ) )
182	181, 141	eqtrd 2505	. . . . . . 7 |- ( ph -> ( 0 A 1 ) = P )
183	92	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 e. ( 0 [,] 1 ) )
184	37, 37, 183	cnmptc 20754	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 1 ) e. ( II Cn II ) )
185	37, 61, 184, 17	cnmpt12f 20758	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) e. ( II Cn C ) )
186	1	cvmlift 30094	. . . . . . . . 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 1293	. . . . . . . 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 2473	. . . . . . . . . 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 5878	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) )
191		oveq1 6315	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 1 ) = ( 0 A 1 ) )
192		eqid 2471	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) )
193		ovex 6336	. . . . . . . . . . . . . 14 |- ( 0 A 1 ) e. _V
194	191, 192, 193	fvmpt 5963	. . . . . . . . . . . . 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 2521	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( 0 A 1 ) )
197	196	eqeq1d 2473	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( ( f ` 0 ) = P <-> ( 0 A 1 ) = P ) )
198	189, 197	anbi12d 725	. . . . . . . . 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 6292	. . . . . . . 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 673	. . . . . . 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 935	. . . . . 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 2517	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
203	19, 1	cnf 20339	. . . . . . 7 |- ( N e. ( II Cn C ) -> N : ( 0 [,] 1 ) --> B )
204	16, 203	syl 17	. . . . . 6 |- ( ph -> N : ( 0 [,] 1 ) --> B )
205	204	feqmptd 5932	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
206	202, 205	eqtr3d 2507	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
207		mpteqb 5979	. . . . 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 6336	. . . . . 6 |- ( s A 1 ) e. _V
209	208	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 1 ) e. _V )
210	207, 209	mprg 2770	. . . 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 201	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( s A 1 ) = ( N ` s ) )
212	211	r19.21bi 2776	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) = ( N ` s ) )
213	177	r19.21bi 2776	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) = ( M ` 0 ) )
214	37, 184, 61, 17	cnmpt12f 20758	. . . . . 6 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) e. ( II Cn C ) )
215		ffvelrn 6035	. . . . . . . 8 |- ( ( M : ( 0 [,] 1 ) --> B /\ 1 e. ( 0 [,] 1 ) ) -> ( M ` 1 ) e. B )
216	83, 92, 215	sylancl 675	. . . . . . 7 |- ( ph -> ( M ` 1 ) e. B )
217		cnconst2 20376	. . . . . . 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 1292	. . . . . 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 684	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
221		fvco3 5957	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( F ` ( A ` <. 1 , s >. ) ) )
223	29	fveq1d 5881	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( K ` <. 1 , s >. ) )
224	222, 223	eqtr3d 2507	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 1 , s >. ) ) = ( K ` <. 1 , s >. ) )
225		df-ov 6311	. . . . . . . . . . . 12 |- ( 1 A s ) = ( A ` <. 1 , s >. )
226	225	fveq2i 5882	. . . . . . . . . . 11 |- ( F ` ( 1 A s ) ) = ( F ` ( A ` <. 1 , s >. ) )
227		df-ov 6311	. . . . . . . . . . 11 |- ( 1 K s ) = ( K ` <. 1 , s >. )
228	224, 226, 227	3eqtr4g 2530	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( 1 K s ) )
229	129	simprd 470	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 K s ) = ( G ` 1 ) )
230	7	simp2d 1043	. . . . . . . . . . . . 13 |- ( ph -> ( F o. M ) = G )
231	230	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. M ) = G )
232	231	fveq1d 5881	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( G ` 1 ) )
233	83	adantr 472	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> M : ( 0 [,] 1 ) --> B )
234		fvco3 5957	. . . . . . . . . . . 12 |- ( ( M : ( 0 [,] 1 ) --> B /\ 1 e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( F ` ( M ` 1 ) ) )
235	233, 92, 234	sylancl 675	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( F ` ( M ` 1 ) ) )
236	232, 235	eqtr3d 2507	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` 1 ) = ( F ` ( M ` 1 ) ) )
237	228, 229, 236	3eqtrd 2509	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( F ` ( M ` 1 ) ) )
238	237	mpteq2dva 4482	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) ) )
239		fconstmpt 4883	. . . . . . . 8 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) )
240	238, 239	syl6eqr 2523	. . . . . . 7 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
241		fovrn 6458	. . . . . . . . . 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 1378	. . . . . . . . 9 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
243	22, 242	sylan 479	. . . . . . . 8 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
244		eqidd 2472	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) )
245		fveq2 5879	. . . . . . . 8 |- ( x = ( 1 A s ) -> ( F ` x ) = ( F ` ( 1 A s ) ) )
246	243, 244, 53, 245	fmptco 6072	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) )
247		fcoconst 6076	. . . . . . . 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 673	. . . . . . 7 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
249	240, 246, 248	3eqtr4d 2515	. . . . . 6 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) )
250		oveq1 6315	. . . . . . . . . 10 |- ( s = 1 -> ( s A 0 ) = ( 1 A 0 ) )
251		fveq2 5879	. . . . . . . . . 10 |- ( s = 1 -> ( M ` s ) = ( M ` 1 ) )
252	250, 251	eqeq12d 2486	. . . . . . . . 9 |- ( s = 1 -> ( ( s A 0 ) = ( M ` s ) <-> ( 1 A 0 ) = ( M ` 1 ) ) )
253	252	rspcv 3132	. . . . . . . 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 64	. . . . . . 7 |- ( ph -> ( 1 A 0 ) = ( M ` 1 ) )
255		oveq2 6316	. . . . . . . . 9 |- ( s = 0 -> ( 1 A s ) = ( 1 A 0 ) )
256		eqid 2471	. . . . . . . . 9 |- ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) )
257		ovex 6336	. . . . . . . . 9 |- ( 1 A 0 ) e. _V
258	255, 256, 257	fvmpt 5963	. . . . . . . 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 5889	. . . . . . . . 9 |- ( M ` 1 ) e. _V
261	260	fvconst2 6136	. . . . . . . 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 2530	. . . . . 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 30078	. . . . 5 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) )
265		fconstmpt 4883	. . . . 5 |- ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) )
266	264, 265	syl6eq 2521	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) ) )
267		mpteqb 5979	. . . . 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 6336	. . . . . 6 |- ( 1 A s ) e. _V
269	268	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( 1 A s ) e. _V )
270	267, 269	mprg 2770	. . . 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 201	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( 1 A s ) = ( M ` 1 ) )
272	271	r19.21bi 2776	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) = ( M ` 1 ) )
273	8, 16, 17, 91, 212, 213, 272	isphtpy2d 22096	1 |- ( ph -> A e. ( M ( PHtpy ` C ) N ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   E!wreu 2758   _Vcvv 3031   {csn 3959   <.cop 3965   U.cuni 4190   |-> cmpt 4454   X. cxp 4837   o. ccom 4843   Fn wfn 5584   -->wf 5585   ` cfv 5589   iota_crio 6269  (class class class)co 6308   0cc0 9557   1c1 9558   [,]cicc 11663   Topctop 19994  TopOnctopon 19995   Cn ccn 20317   Conccon 20503  𝑛Locally cnlly 20557   tX ctx 20652   IIcii 21985   Htpy chtpy 22076   PHtpycphtpy 22077   CovMap ccvm 30050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-ec 7383  df-map 7492  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-cn 20320  df-cnp 20321  df-cmp 20479  df-con 20504  df-lly 20558  df-nlly 20559  df-tx 20654  df-hmeo 20847  df-xms 21413  df-ms 21414  df-tms 21415  df-ii 21987  df-htpy 22079  df-phtpy 22080  df-phtpc 22101  df-pcon 30016  df-scon 30017  df-cvm 30051

This theorem is referenced by:  cvmliftpht  30113