Theorem cvmliftphtlem 28937

Description: Lemma for cvmliftpht 28938. (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 28921	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
8	7	simp1d 1008	. 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 21610	. . . . . 6 |- ( ph -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
13	12	simpld 459	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
14	6, 13	eqtrd 2498	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
15	1, 9, 3, 10, 5, 14	cvmliftiota 28921	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
16	15	simp1d 1008	. 2 |- ( ph -> N e. ( II Cn C ) )
17		cvmliftphtlem.a	. 2 |- ( ph -> A e. ( ( II tX II ) Cn C ) )
18		iitop 21509	. . . . . . . . . . . . . . . 16 |- II e. Top
19		iiuni 21510	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) = U. II
20	18, 18, 19, 19	txunii 20219	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
21	20, 1	cnf 19873	. . . . . . . . . . . . . 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 11663	. . . . . . . . . . . . . 14 |- 0 e. ( 0 [,] 1 )
24		opelxpi 5040	. . . . . . . . . . . . . 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 5950	. . . . . . . . . . . . 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 5874	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 0 >. ) = ( K ` <. s , 0 >. ) )
31	27, 30	eqtr3d 2500	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 0 >. ) ) = ( K ` <. s , 0 >. ) )
32		df-ov 6299	. . . . . . . . . . . 12 |- ( s A 0 ) = ( A ` <. s , 0 >. )
33	32	fveq2i 5875	. . . . . . . . . . 11 |- ( F ` ( s A 0 ) ) = ( F ` ( A ` <. s , 0 >. ) )
34		df-ov 6299	. . . . . . . . . . 11 |- ( s K 0 ) = ( K ` <. s , 0 >. )
35	31, 33, 34	3eqtr4g 2523	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( s K 0 ) )
36		iitopon 21508	. . . . . . . . . . . . 13 |- II e. (TopOn ` ( 0 [,] 1 ) )
37	36	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
38	4, 10	phtpyhtpy 21607	. . . . . . . . . . . . 13 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( G ( II Htpy J ) H ) )
39	38, 11	sseldd 3500	. . . . . . . . . . . 12 |- ( ph -> K e. ( G ( II Htpy J ) H ) )
40	37, 4, 10, 39	htpyi 21599	. . . . . . . . . . 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 2498	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( G ` s ) )
43	42	mpteq2dva 4543	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
44		fovrn 6444	. . . . . . . . . . 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 1313	. . . . . . . . . 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 2458	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
48		cvmcn 28882	. . . . . . . . . . . 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 2457	. . . . . . . . . . . 12 |- U. J = U. J
51	1, 50	cnf 19873	. . . . . . . . . . 11 |- ( F e. ( C Cn J ) -> F : B --> U. J )
52	49, 51	syl 16	. . . . . . . . . 10 |- ( ph -> F : B --> U. J )
53	52	feqmptd 5926	. . . . . . . . 9 |- ( ph -> F = ( x e. B |-> ( F ` x ) ) )
54		fveq2 5872	. . . . . . . . 9 |- ( x = ( s A 0 ) -> ( F ` x ) = ( F ` ( s A 0 ) ) )
55	46, 47, 53, 54	fmptco 6065	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) )
56	19, 50	cnf 19873	. . . . . . . . . 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 5926	. . . . . . . 8 |- ( ph -> G = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
59	43, 55, 58	3eqtr4d 2508	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G )
60		cvmliftphtlem.0	. . . . . . 7 |- ( ph -> ( 0 A 0 ) = P )
61	37	cnmptid 20287	. . . . . . . . 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 20288	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
64	37, 61, 63, 17	cnmpt12f 20292	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) e. ( II Cn C ) )
65	1	cvmlift 28919	. . . . . . . . 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 1229	. . . . . . . 8 |- ( ph -> E! f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
67		coeq2 5171	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( F o. f ) = ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) )
68	67	eqeq1d 2459	. . . . . . . . . 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 5871	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) )
70		oveq1 6303	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 0 ) = ( 0 A 0 ) )
71		eqid 2457	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) )
72		ovex 6324	. . . . . . . . . . . . . 14 |- ( 0 A 0 ) e. _V
73	70, 71, 72	fvmpt 5956	. . . . . . . . . . . . 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 2514	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( 0 A 0 ) )
76	75	eqeq1d 2459	. . . . . . . . . 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 6280	. . . . . . . 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 921	. . . . . 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 2510	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
82	19, 1	cnf 19873	. . . . . . 7 |- ( M e. ( II Cn C ) -> M : ( 0 [,] 1 ) --> B )
83	8, 82	syl 16	. . . . . 6 |- ( ph -> M : ( 0 [,] 1 ) --> B )
84	83	feqmptd 5926	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
85	81, 84	eqtr3d 2500	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
86		mpteqb 5971	. . . . 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 6324	. . . . . 6 |- ( s A 0 ) e. _V
88	87	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 0 ) e. _V )
89	86, 88	mprg 2820	. . . 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 2826	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) = ( M ` s ) )
92		1elunit 11664	. . . . . . . . . . . . . 14 |- 1 e. ( 0 [,] 1 )
93		opelxpi 5040	. . . . . . . . . . . . . 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 5950	. . . . . . . . . . . . 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 5874	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( K ` <. s , 1 >. ) )
98	96, 97	eqtr3d 2500	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 1 >. ) ) = ( K ` <. s , 1 >. ) )
99		df-ov 6299	. . . . . . . . . . . 12 |- ( s A 1 ) = ( A ` <. s , 1 >. )
100	99	fveq2i 5875	. . . . . . . . . . 11 |- ( F ` ( s A 1 ) ) = ( F ` ( A ` <. s , 1 >. ) )
101		df-ov 6299	. . . . . . . . . . 11 |- ( s K 1 ) = ( K ` <. s , 1 >. )
102	98, 100, 101	3eqtr4g 2523	. . . . . . . . . 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 2498	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( H ` s ) )
105	104	mpteq2dva 4543	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
106		fovrn 6444	. . . . . . . . . . 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 1313	. . . . . . . . . 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 2458	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
110		fveq2 5872	. . . . . . . . 9 |- ( x = ( s A 1 ) -> ( F ` x ) = ( F ` ( s A 1 ) ) )
111	108, 109, 53, 110	fmptco 6065	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) )
112	19, 50	cnf 19873	. . . . . . . . . 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 5926	. . . . . . . 8 |- ( ph -> H = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
115	105, 111, 114	3eqtr4d 2508	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H )
116		iicon 21516	. . . . . . . . . . . . 13 |- II e. Con
117	116	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. Con )
118		iinllycon 28874	. . . . . . . . . . . . 13 |- II e. 𝑛Locally Con
119	118	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. 𝑛Locally Con )
120	37, 63, 61, 17	cnmpt12f 20292	. . . . . . . . . . . 12 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) e. ( II Cn C ) )
121		cvmtop1 28880	. . . . . . . . . . . . . . 15 |- ( F e. ( C CovMap J ) -> C e. Top )
122	3, 121	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> C e. Top )
123	1	toptopon 19560	. . . . . . . . . . . . . 14 |- ( C e. Top <-> C e. (TopOn ` B ) )
124	122, 123	sylib 196	. . . . . . . . . . . . 13 |- ( ph -> C e. (TopOn ` B ) )
125		ffvelrn 6030	. . . . . . . . . . . . . 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 19910	. . . . . . . . . . . . 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 1228	. . . . . . . . . . . 12 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) e. ( II Cn C ) )
129	4, 10, 11	phtpyi 21609	. . . . . . . . . . . . . . . . 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 5040	. . . . . . . . . . . . . . . . . . . 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 5950	. . . . . . . . . . . . . . . . . . 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 5874	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( K ` <. 0 , s >. ) )
136	134, 135	eqtr3d 2500	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 0 , s >. ) ) = ( K ` <. 0 , s >. ) )
137		df-ov 6299	. . . . . . . . . . . . . . . . . 18 |- ( 0 A s ) = ( A ` <. 0 , s >. )
138	137	fveq2i 5875	. . . . . . . . . . . . . . . . 17 |- ( F ` ( 0 A s ) ) = ( F ` ( A ` <. 0 , s >. ) )
139		df-ov 6299	. . . . . . . . . . . . . . . . 17 |- ( 0 K s ) = ( K ` <. 0 , s >. )
140	136, 138, 139	3eqtr4g 2523	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( 0 K s ) )
141	7	simp3d 1010	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( M ` 0 ) = P )
142	141	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( M ` 0 ) = P )
143	142	fveq2d 5876	. . . . . . . . . . . . . . . . 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 2498	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( G ` 0 ) )
146	130, 140, 145	3eqtr4d 2508	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( F ` ( M ` 0 ) ) )
147	146	mpteq2dva 4543	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) ) )
148		fconstmpt 5052	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) )
149	147, 148	syl6eqr 2516	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
150		fovrn 6444	. . . . . . . . . . . . . . . 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 1312	. . . . . . . . . . . . . . 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 2458	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) )
154		fveq2 5872	. . . . . . . . . . . . . 14 |- ( x = ( 0 A s ) -> ( F ` x ) = ( F ` ( 0 A s ) ) )
155	152, 153, 53, 154	fmptco 6065	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) )
156		ffn 5737	. . . . . . . . . . . . . . 15 |- ( F : B --> U. J -> F Fn B )
157	52, 156	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> F Fn B )
158		fcoconst 6069	. . . . . . . . . . . . . 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 2508	. . . . . . . . . . . 12 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) )
161	60, 141	eqtr4d 2501	. . . . . . . . . . . . 13 |- ( ph -> ( 0 A 0 ) = ( M ` 0 ) )
162		oveq2 6304	. . . . . . . . . . . . . . 15 |- ( s = 0 -> ( 0 A s ) = ( 0 A 0 ) )
163		eqid 2457	. . . . . . . . . . . . . . 15 |- ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) )
164	162, 163, 72	fvmpt 5956	. . . . . . . . . . . . . 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 5882	. . . . . . . . . . . . . . 15 |- ( M ` 0 ) e. _V
167	166	fvconst2 6128	. . . . . . . . . . . . . 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 2523	. . . . . . . . . . . 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 28903	. . . . . . . . . . 11 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) )
171		fconstmpt 5052	. . . . . . . . . . 11 |- ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) )
172	170, 171	syl6eq 2514	. . . . . . . . . 10 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) ) )
173		mpteqb 5971	. . . . . . . . . . 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 6324	. . . . . . . . . . . 12 |- ( 0 A s ) e. _V
175	174	a1i 11	. . . . . . . . . . 11 |- ( s e. ( 0 [,] 1 ) -> ( 0 A s ) e. _V )
176	173, 175	mprg 2820	. . . . . . . . . 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 6304	. . . . . . . . . . 11 |- ( s = 1 -> ( 0 A s ) = ( 0 A 1 ) )
179	178	eqeq1d 2459	. . . . . . . . . 10 |- ( s = 1 -> ( ( 0 A s ) = ( M ` 0 ) <-> ( 0 A 1 ) = ( M ` 0 ) ) )
180	179	rspcv 3206	. . . . . . . . 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 2498	. . . . . . 7 |- ( ph -> ( 0 A 1 ) = P )
183	92	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 e. ( 0 [,] 1 ) )
184	37, 37, 183	cnmptc 20288	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 1 ) e. ( II Cn II ) )
185	37, 61, 184, 17	cnmpt12f 20292	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) e. ( II Cn C ) )
186	1	cvmlift 28919	. . . . . . . . 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 1229	. . . . . . . 8 |- ( ph -> E! f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
188		coeq2 5171	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( F o. f ) = ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) )
189	188	eqeq1d 2459	. . . . . . . . . 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 5871	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) )
191		oveq1 6303	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 1 ) = ( 0 A 1 ) )
192		eqid 2457	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) )
193		ovex 6324	. . . . . . . . . . . . . 14 |- ( 0 A 1 ) e. _V
194	191, 192, 193	fvmpt 5956	. . . . . . . . . . . . 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 2514	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( 0 A 1 ) )
197	196	eqeq1d 2459	. . . . . . . . . 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 6280	. . . . . . . 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 921	. . . . . 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 2510	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
203	19, 1	cnf 19873	. . . . . . 7 |- ( N e. ( II Cn C ) -> N : ( 0 [,] 1 ) --> B )
204	16, 203	syl 16	. . . . . 6 |- ( ph -> N : ( 0 [,] 1 ) --> B )
205	204	feqmptd 5926	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
206	202, 205	eqtr3d 2500	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
207		mpteqb 5971	. . . . 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 6324	. . . . . 6 |- ( s A 1 ) e. _V
209	208	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 1 ) e. _V )
210	207, 209	mprg 2820	. . . 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 2826	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) = ( N ` s ) )
213	177	r19.21bi 2826	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) = ( M ` 0 ) )
214	37, 184, 61, 17	cnmpt12f 20292	. . . . . 6 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) e. ( II Cn C ) )
215		ffvelrn 6030	. . . . . . . 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 19910	. . . . . . 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 1228	. . . . . 6 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) e. ( II Cn C ) )
219		opelxpi 5040	. . . . . . . . . . . . . 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 5950	. . . . . . . . . . . . 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 5874	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( K ` <. 1 , s >. ) )
224	222, 223	eqtr3d 2500	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 1 , s >. ) ) = ( K ` <. 1 , s >. ) )
225		df-ov 6299	. . . . . . . . . . . 12 |- ( 1 A s ) = ( A ` <. 1 , s >. )
226	225	fveq2i 5875	. . . . . . . . . . 11 |- ( F ` ( 1 A s ) ) = ( F ` ( A ` <. 1 , s >. ) )
227		df-ov 6299	. . . . . . . . . . 11 |- ( 1 K s ) = ( K ` <. 1 , s >. )
228	224, 226, 227	3eqtr4g 2523	. . . . . . . . . 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 1009	. . . . . . . . . . . . 13 |- ( ph -> ( F o. M ) = G )
231	230	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. M ) = G )
232	231	fveq1d 5874	. . . . . . . . . . 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 5950	. . . . . . . . . . . 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 2500	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` 1 ) = ( F ` ( M ` 1 ) ) )
237	228, 229, 236	3eqtrd 2502	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( F ` ( M ` 1 ) ) )
238	237	mpteq2dva 4543	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) ) )
239		fconstmpt 5052	. . . . . . . 8 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) )
240	238, 239	syl6eqr 2516	. . . . . . 7 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
241		fovrn 6444	. . . . . . . . . 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 1312	. . . . . . . . 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 2458	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) )
245		fveq2 5872	. . . . . . . 8 |- ( x = ( 1 A s ) -> ( F ` x ) = ( F ` ( 1 A s ) ) )
246	243, 244, 53, 245	fmptco 6065	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) )
247		fcoconst 6069	. . . . . . . 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 2508	. . . . . 6 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) )
250		oveq1 6303	. . . . . . . . . 10 |- ( s = 1 -> ( s A 0 ) = ( 1 A 0 ) )
251		fveq2 5872	. . . . . . . . . 10 |- ( s = 1 -> ( M ` s ) = ( M ` 1 ) )
252	250, 251	eqeq12d 2479	. . . . . . . . 9 |- ( s = 1 -> ( ( s A 0 ) = ( M ` s ) <-> ( 1 A 0 ) = ( M ` 1 ) ) )
253	252	rspcv 3206	. . . . . . . 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 6304	. . . . . . . . 9 |- ( s = 0 -> ( 1 A s ) = ( 1 A 0 ) )
256		eqid 2457	. . . . . . . . 9 |- ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) )
257		ovex 6324	. . . . . . . . 9 |- ( 1 A 0 ) e. _V
258	255, 256, 257	fvmpt 5956	. . . . . . . 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 5882	. . . . . . . . 9 |- ( M ` 1 ) e. _V
261	260	fvconst2 6128	. . . . . . . 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 2523	. . . . . 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 28903	. . . . 5 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) )
265		fconstmpt 5052	. . . . 5 |- ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) )
266	264, 265	syl6eq 2514	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) ) )
267		mpteqb 5971	. . . . 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 6324	. . . . . 6 |- ( 1 A s ) e. _V
269	268	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( 1 A s ) e. _V )
270	267, 269	mprg 2820	. . . 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 2826	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) = ( M ` 1 ) )
273	8, 16, 17, 91, 212, 213, 272	isphtpy2d 21612	1 |- ( ph -> A e. ( M ( PHtpy ` C ) N ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   E!wreu 2809   _Vcvv 3109   {csn 4032   <.cop 4038   U.cuni 4251   |-> cmpt 4515   X. cxp 5006   o. ccom 5012   Fn wfn 5589   -->wf 5590   ` cfv 5594   iota_crio 6257  (class class class)co 6296   0cc0 9509   1c1 9510   [,]cicc 11557   Topctop 19520  TopOnctopon 19521   Cn ccn 19851   Conccon 20037  𝑛Locally cnlly 20091   tX ctx 20186   IIcii 21504   Htpy chtpy 21592   PHtpycphtpy 21593   CovMap ccvm 28875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-ec 7331  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-sum 13520  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-cn 19854  df-cnp 19855  df-cmp 20013  df-con 20038  df-lly 20092  df-nlly 20093  df-tx 20188  df-hmeo 20381  df-xms 20948  df-ms 20949  df-tms 20950  df-ii 21506  df-htpy 21595  df-phtpy 21596  df-phtpc 21617  df-pcon 28841  df-scon 28842  df-cvm 28876

This theorem is referenced by:  cvmliftpht  28938