Theorem cvmliftphtlem 30033

Description: Lemma for cvmliftpht 30034. (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 30017	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
8	7	simp1d 1019	. 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 22009	. . . . . 6 |- ( ph -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
13	12	simpld 461	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
14	6, 13	eqtrd 2484	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
15	1, 9, 3, 10, 5, 14	cvmliftiota 30017	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
16	15	simp1d 1019	. 2 |- ( ph -> N e. ( II Cn C ) )
17		cvmliftphtlem.a	. 2 |- ( ph -> A e. ( ( II tX II ) Cn C ) )
18		iitop 21905	. . . . . . . . . . . . . . . 16 |- II e. Top
19		iiuni 21906	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) = U. II
20	18, 18, 19, 19	txunii 20601	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) = U. ( II tX II )
21	20, 1	cnf 20255	. . . . . . . . . . . . . 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 11747	. . . . . . . . . . . . . 14 |- 0 e. ( 0 [,] 1 )
24		opelxpi 4865	. . . . . . . . . . . . . 14 |- ( ( s e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) ) -> <. s , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
25	23, 24	mpan2 676	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. s , 0 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
26		fvco3 5940	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 467	. . . . . . . . . . . . 13 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. A ) = K )
30	29	fveq1d 5865	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 0 >. ) = ( K ` <. s , 0 >. ) )
31	27, 30	eqtr3d 2486	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 0 >. ) ) = ( K ` <. s , 0 >. ) )
32		df-ov 6291	. . . . . . . . . . . 12 |- ( s A 0 ) = ( A ` <. s , 0 >. )
33	32	fveq2i 5866	. . . . . . . . . . 11 |- ( F ` ( s A 0 ) ) = ( F ` ( A ` <. s , 0 >. ) )
34		df-ov 6291	. . . . . . . . . . 11 |- ( s K 0 ) = ( K ` <. s , 0 >. )
35	31, 33, 34	3eqtr4g 2509	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( s K 0 ) )
36		iitopon 21904	. . . . . . . . . . . . 13 |- II e. (TopOn ` ( 0 [,] 1 ) )
37	36	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
38	4, 10	phtpyhtpy 22006	. . . . . . . . . . . . 13 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( G ( II Htpy J ) H ) )
39	38, 11	sseldd 3432	. . . . . . . . . . . 12 |- ( ph -> K e. ( G ( II Htpy J ) H ) )
40	37, 4, 10, 39	htpyi 21998	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( s K 0 ) = ( G ` s ) /\ ( s K 1 ) = ( H ` s ) ) )
41	40	simpld 461	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s K 0 ) = ( G ` s ) )
42	35, 41	eqtrd 2484	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 0 ) ) = ( G ` s ) )
43	42	mpteq2dva 4488	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
44		fovrn 6436	. . . . . . . . . . 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 1352	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
46	22, 45	sylan 474	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) e. B )
47		eqidd 2451	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
48		cvmcn 29978	. . . . . . . . . . . 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 2450	. . . . . . . . . . . 12 |- U. J = U. J
51	1, 50	cnf 20255	. . . . . . . . . . 11 |- ( F e. ( C Cn J ) -> F : B --> U. J )
52	49, 51	syl 17	. . . . . . . . . 10 |- ( ph -> F : B --> U. J )
53	52	feqmptd 5916	. . . . . . . . 9 |- ( ph -> F = ( x e. B |-> ( F ` x ) ) )
54		fveq2 5863	. . . . . . . . 9 |- ( x = ( s A 0 ) -> ( F ` x ) = ( F ` ( s A 0 ) ) )
55	46, 47, 53, 54	fmptco 6054	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 0 ) ) ) )
56	19, 50	cnf 20255	. . . . . . . . . 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 5916	. . . . . . . 8 |- ( ph -> G = ( s e. ( 0 [,] 1 ) |-> ( G ` s ) ) )
59	43, 55, 58	3eqtr4d 2494	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) = G )
60		cvmliftphtlem.0	. . . . . . 7 |- ( ph -> ( 0 A 0 ) = P )
61	37	cnmptid 20669	. . . . . . . . 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 20670	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 0 ) e. ( II Cn II ) )
64	37, 61, 63, 17	cnmpt12f 20674	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) e. ( II Cn C ) )
65	1	cvmlift 30015	. . . . . . . . 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 1268	. . . . . . . 8 |- ( ph -> E! f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
67		coeq2 4992	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( F o. f ) = ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ) )
68	67	eqeq1d 2452	. . . . . . . . . 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 5862	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) ` 0 ) )
70		oveq1 6295	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 0 ) = ( 0 A 0 ) )
71		eqid 2450	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) )
72		ovex 6316	. . . . . . . . . . . . . 14 |- ( 0 A 0 ) e. _V
73	70, 71, 72	fvmpt 5946	. . . . . . . . . . . . 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 2500	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( f ` 0 ) = ( 0 A 0 ) )
76	75	eqeq1d 2452	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) -> ( ( f ` 0 ) = P <-> ( 0 A 0 ) = P ) )
77	68, 76	anbi12d 716	. . . . . . . . 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 6272	. . . . . . . 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 666	. . . . . . 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 931	. . . . . 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 2496	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) )
82	19, 1	cnf 20255	. . . . . . 7 |- ( M e. ( II Cn C ) -> M : ( 0 [,] 1 ) --> B )
83	8, 82	syl 17	. . . . . 6 |- ( ph -> M : ( 0 [,] 1 ) --> B )
84	83	feqmptd 5916	. . . . 5 |- ( ph -> M = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
85	81, 84	eqtr3d 2486	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 0 ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` s ) ) )
86		mpteqb 5962	. . . . 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 6316	. . . . . 6 |- ( s A 0 ) e. _V
88	87	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 0 ) e. _V )
89	86, 88	mprg 2750	. . . 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 200	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( s A 0 ) = ( M ` s ) )
91	90	r19.21bi 2756	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 0 ) = ( M ` s ) )
92		1elunit 11748	. . . . . . . . . . . . . 14 |- 1 e. ( 0 [,] 1 )
93		opelxpi 4865	. . . . . . . . . . . . . 14 |- ( ( s e. ( 0 [,] 1 ) /\ 1 e. ( 0 [,] 1 ) ) -> <. s , 1 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
94	92, 93	mpan2 676	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. s , 1 >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
95		fvco3 5940	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( F ` ( A ` <. s , 1 >. ) ) )
97	29	fveq1d 5865	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. s , 1 >. ) = ( K ` <. s , 1 >. ) )
98	96, 97	eqtr3d 2486	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. s , 1 >. ) ) = ( K ` <. s , 1 >. ) )
99		df-ov 6291	. . . . . . . . . . . 12 |- ( s A 1 ) = ( A ` <. s , 1 >. )
100	99	fveq2i 5866	. . . . . . . . . . 11 |- ( F ` ( s A 1 ) ) = ( F ` ( A ` <. s , 1 >. ) )
101		df-ov 6291	. . . . . . . . . . 11 |- ( s K 1 ) = ( K ` <. s , 1 >. )
102	98, 100, 101	3eqtr4g 2509	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( s K 1 ) )
103	40	simprd 465	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s K 1 ) = ( H ` s ) )
104	102, 103	eqtrd 2484	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( s A 1 ) ) = ( H ` s ) )
105	104	mpteq2dva 4488	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
106		fovrn 6436	. . . . . . . . . . 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 1352	. . . . . . . . . 10 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
108	22, 107	sylan 474	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) e. B )
109		eqidd 2451	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
110		fveq2 5863	. . . . . . . . 9 |- ( x = ( s A 1 ) -> ( F ` x ) = ( F ` ( s A 1 ) ) )
111	108, 109, 53, 110	fmptco 6054	. . . . . . . 8 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( s A 1 ) ) ) )
112	19, 50	cnf 20255	. . . . . . . . . 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 5916	. . . . . . . 8 |- ( ph -> H = ( s e. ( 0 [,] 1 ) |-> ( H ` s ) ) )
115	105, 111, 114	3eqtr4d 2494	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) = H )
116		iicon 21912	. . . . . . . . . . . . 13 |- II e. Con
117	116	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. Con )
118		iinllycon 29970	. . . . . . . . . . . . 13 |- II e. 𝑛Locally Con
119	118	a1i 11	. . . . . . . . . . . 12 |- ( ph -> II e. 𝑛Locally Con )
120	37, 63, 61, 17	cnmpt12f 20674	. . . . . . . . . . . 12 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) e. ( II Cn C ) )
121		cvmtop1 29976	. . . . . . . . . . . . . . 15 |- ( F e. ( C CovMap J ) -> C e. Top )
122	3, 121	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> C e. Top )
123	1	toptopon 19941	. . . . . . . . . . . . . 14 |- ( C e. Top <-> C e. (TopOn ` B ) )
124	122, 123	sylib 200	. . . . . . . . . . . . 13 |- ( ph -> C e. (TopOn ` B ) )
125		ffvelrn 6018	. . . . . . . . . . . . . 14 |- ( ( M : ( 0 [,] 1 ) --> B /\ 0 e. ( 0 [,] 1 ) ) -> ( M ` 0 ) e. B )
126	83, 23, 125	sylancl 667	. . . . . . . . . . . . 13 |- ( ph -> ( M ` 0 ) e. B )
127		cnconst2 20292	. . . . . . . . . . . . 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 1267	. . . . . . . . . . . 12 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) e. ( II Cn C ) )
129	4, 10, 11	phtpyi 22008	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 K s ) = ( G ` 0 ) /\ ( 1 K s ) = ( G ` 1 ) ) )
130	129	simpld 461	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 K s ) = ( G ` 0 ) )
131		opelxpi 4865	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 0 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
132	23, 131	mpan 675	. . . . . . . . . . . . . . . . . . 19 |- ( s e. ( 0 [,] 1 ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
133		fvco3 5940	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( F ` ( A ` <. 0 , s >. ) ) )
135	29	fveq1d 5865	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 0 , s >. ) = ( K ` <. 0 , s >. ) )
136	134, 135	eqtr3d 2486	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 0 , s >. ) ) = ( K ` <. 0 , s >. ) )
137		df-ov 6291	. . . . . . . . . . . . . . . . . 18 |- ( 0 A s ) = ( A ` <. 0 , s >. )
138	137	fveq2i 5866	. . . . . . . . . . . . . . . . 17 |- ( F ` ( 0 A s ) ) = ( F ` ( A ` <. 0 , s >. ) )
139		df-ov 6291	. . . . . . . . . . . . . . . . 17 |- ( 0 K s ) = ( K ` <. 0 , s >. )
140	136, 138, 139	3eqtr4g 2509	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( 0 K s ) )
141	7	simp3d 1021	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( M ` 0 ) = P )
142	141	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( M ` 0 ) = P )
143	142	fveq2d 5867	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( F ` P ) )
144	6	adantr 467	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` P ) = ( G ` 0 ) )
145	143, 144	eqtrd 2484	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( M ` 0 ) ) = ( G ` 0 ) )
146	130, 140, 145	3eqtr4d 2494	. . . . . . . . . . . . . . 15 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 0 A s ) ) = ( F ` ( M ` 0 ) ) )
147	146	mpteq2dva 4488	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) ) )
148		fconstmpt 4877	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 0 ) ) )
149	147, 148	syl6eqr 2502	. . . . . . . . . . . . 13 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
150		fovrn 6436	. . . . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . . . 15 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
152	22, 151	sylan 474	. . . . . . . . . . . . . 14 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) e. B )
153		eqidd 2451	. . . . . . . . . . . . . 14 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) )
154		fveq2 5863	. . . . . . . . . . . . . 14 |- ( x = ( 0 A s ) -> ( F ` x ) = ( F ` ( 0 A s ) ) )
155	152, 153, 53, 154	fmptco 6054	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 0 A s ) ) ) )
156		ffn 5726	. . . . . . . . . . . . . . 15 |- ( F : B --> U. J -> F Fn B )
157	52, 156	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> F Fn B )
158		fcoconst 6058	. . . . . . . . . . . . . 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 666	. . . . . . . . . . . . 13 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 0 ) ) } ) )
160	149, 155, 159	3eqtr4d 2494	. . . . . . . . . . . 12 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) ) )
161	60, 141	eqtr4d 2487	. . . . . . . . . . . . 13 |- ( ph -> ( 0 A 0 ) = ( M ` 0 ) )
162		oveq2 6296	. . . . . . . . . . . . . . 15 |- ( s = 0 -> ( 0 A s ) = ( 0 A 0 ) )
163		eqid 2450	. . . . . . . . . . . . . . 15 |- ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) )
164	162, 163, 72	fvmpt 5946	. . . . . . . . . . . . . 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 5873	. . . . . . . . . . . . . . 15 |- ( M ` 0 ) e. _V
167	166	fvconst2 6118	. . . . . . . . . . . . . 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 2509	. . . . . . . . . . . 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 29999	. . . . . . . . . . 11 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) )
171		fconstmpt 4877	. . . . . . . . . . 11 |- ( ( 0 [,] 1 ) X. { ( M ` 0 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) )
172	170, 171	syl6eq 2500	. . . . . . . . . 10 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 0 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 0 ) ) )
173		mpteqb 5962	. . . . . . . . . . 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 6316	. . . . . . . . . . . 12 |- ( 0 A s ) e. _V
175	174	a1i 11	. . . . . . . . . . 11 |- ( s e. ( 0 [,] 1 ) -> ( 0 A s ) e. _V )
176	173, 175	mprg 2750	. . . . . . . . . 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 200	. . . . . . . . 9 |- ( ph -> A. s e. ( 0 [,] 1 ) ( 0 A s ) = ( M ` 0 ) )
178		oveq2 6296	. . . . . . . . . . 11 |- ( s = 1 -> ( 0 A s ) = ( 0 A 1 ) )
179	178	eqeq1d 2452	. . . . . . . . . 10 |- ( s = 1 -> ( ( 0 A s ) = ( M ` 0 ) <-> ( 0 A 1 ) = ( M ` 0 ) ) )
180	179	rspcv 3145	. . . . . . . . 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 65	. . . . . . . 8 |- ( ph -> ( 0 A 1 ) = ( M ` 0 ) )
182	181, 141	eqtrd 2484	. . . . . . 7 |- ( ph -> ( 0 A 1 ) = P )
183	92	a1i 11	. . . . . . . . . 10 |- ( ph -> 1 e. ( 0 [,] 1 ) )
184	37, 37, 183	cnmptc 20670	. . . . . . . . 9 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> 1 ) e. ( II Cn II ) )
185	37, 61, 184, 17	cnmpt12f 20674	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) e. ( II Cn C ) )
186	1	cvmlift 30015	. . . . . . . . 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 1268	. . . . . . . 8 |- ( ph -> E! f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
188		coeq2 4992	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( F o. f ) = ( F o. ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ) )
189	188	eqeq1d 2452	. . . . . . . . . 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 5862	. . . . . . . . . . . 12 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) ` 0 ) )
191		oveq1 6295	. . . . . . . . . . . . . 14 |- ( s = 0 -> ( s A 1 ) = ( 0 A 1 ) )
192		eqid 2450	. . . . . . . . . . . . . 14 |- ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) )
193		ovex 6316	. . . . . . . . . . . . . 14 |- ( 0 A 1 ) e. _V
194	191, 192, 193	fvmpt 5946	. . . . . . . . . . . . 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 2500	. . . . . . . . . . 11 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( f ` 0 ) = ( 0 A 1 ) )
197	196	eqeq1d 2452	. . . . . . . . . 10 |- ( f = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) -> ( ( f ` 0 ) = P <-> ( 0 A 1 ) = P ) )
198	189, 197	anbi12d 716	. . . . . . . . 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 6272	. . . . . . . 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 666	. . . . . . 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 931	. . . . . 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 2496	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) )
203	19, 1	cnf 20255	. . . . . . 7 |- ( N e. ( II Cn C ) -> N : ( 0 [,] 1 ) --> B )
204	16, 203	syl 17	. . . . . 6 |- ( ph -> N : ( 0 [,] 1 ) --> B )
205	204	feqmptd 5916	. . . . 5 |- ( ph -> N = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
206	202, 205	eqtr3d 2486	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( s A 1 ) ) = ( s e. ( 0 [,] 1 ) |-> ( N ` s ) ) )
207		mpteqb 5962	. . . . 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 6316	. . . . . 6 |- ( s A 1 ) e. _V
209	208	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( s A 1 ) e. _V )
210	207, 209	mprg 2750	. . . 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 200	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( s A 1 ) = ( N ` s ) )
212	211	r19.21bi 2756	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( s A 1 ) = ( N ` s ) )
213	177	r19.21bi 2756	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 A s ) = ( M ` 0 ) )
214	37, 184, 61, 17	cnmpt12f 20674	. . . . . 6 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) e. ( II Cn C ) )
215		ffvelrn 6018	. . . . . . . 8 |- ( ( M : ( 0 [,] 1 ) --> B /\ 1 e. ( 0 [,] 1 ) ) -> ( M ` 1 ) e. B )
216	83, 92, 215	sylancl 667	. . . . . . 7 |- ( ph -> ( M ` 1 ) e. B )
217		cnconst2 20292	. . . . . . 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 1267	. . . . . 6 |- ( ph -> ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) e. ( II Cn C ) )
219		opelxpi 4865	. . . . . . . . . . . . . 14 |- ( ( 1 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
220	92, 219	mpan 675	. . . . . . . . . . . . 13 |- ( s e. ( 0 [,] 1 ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
221		fvco3 5940	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( F ` ( A ` <. 1 , s >. ) ) )
223	29	fveq1d 5865	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. A ) ` <. 1 , s >. ) = ( K ` <. 1 , s >. ) )
224	222, 223	eqtr3d 2486	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( A ` <. 1 , s >. ) ) = ( K ` <. 1 , s >. ) )
225		df-ov 6291	. . . . . . . . . . . 12 |- ( 1 A s ) = ( A ` <. 1 , s >. )
226	225	fveq2i 5866	. . . . . . . . . . 11 |- ( F ` ( 1 A s ) ) = ( F ` ( A ` <. 1 , s >. ) )
227		df-ov 6291	. . . . . . . . . . 11 |- ( 1 K s ) = ( K ` <. 1 , s >. )
228	224, 226, 227	3eqtr4g 2509	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( 1 K s ) )
229	129	simprd 465	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 K s ) = ( G ` 1 ) )
230	7	simp2d 1020	. . . . . . . . . . . . 13 |- ( ph -> ( F o. M ) = G )
231	230	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F o. M ) = G )
232	231	fveq1d 5865	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( G ` 1 ) )
233	83	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> M : ( 0 [,] 1 ) --> B )
234		fvco3 5940	. . . . . . . . . . . 12 |- ( ( M : ( 0 [,] 1 ) --> B /\ 1 e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( F ` ( M ` 1 ) ) )
235	233, 92, 234	sylancl 667	. . . . . . . . . . 11 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( F o. M ) ` 1 ) = ( F ` ( M ` 1 ) ) )
236	232, 235	eqtr3d 2486	. . . . . . . . . 10 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( G ` 1 ) = ( F ` ( M ` 1 ) ) )
237	228, 229, 236	3eqtrd 2488	. . . . . . . . 9 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( F ` ( 1 A s ) ) = ( F ` ( M ` 1 ) ) )
238	237	mpteq2dva 4488	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) ) )
239		fconstmpt 4877	. . . . . . . 8 |- ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( M ` 1 ) ) )
240	238, 239	syl6eqr 2502	. . . . . . 7 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
241		fovrn 6436	. . . . . . . . . 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 1351	. . . . . . . . 9 |- ( ( A : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> B /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
243	22, 242	sylan 474	. . . . . . . 8 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) e. B )
244		eqidd 2451	. . . . . . . 8 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) )
245		fveq2 5863	. . . . . . . 8 |- ( x = ( 1 A s ) -> ( F ` x ) = ( F ` ( 1 A s ) ) )
246	243, 244, 53, 245	fmptco 6054	. . . . . . 7 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( s e. ( 0 [,] 1 ) |-> ( F ` ( 1 A s ) ) ) )
247		fcoconst 6058	. . . . . . . 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 666	. . . . . . 7 |- ( ph -> ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) = ( ( 0 [,] 1 ) X. { ( F ` ( M ` 1 ) ) } ) )
249	240, 246, 248	3eqtr4d 2494	. . . . . 6 |- ( ph -> ( F o. ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) ) = ( F o. ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) ) )
250		oveq1 6295	. . . . . . . . . 10 |- ( s = 1 -> ( s A 0 ) = ( 1 A 0 ) )
251		fveq2 5863	. . . . . . . . . 10 |- ( s = 1 -> ( M ` s ) = ( M ` 1 ) )
252	250, 251	eqeq12d 2465	. . . . . . . . 9 |- ( s = 1 -> ( ( s A 0 ) = ( M ` s ) <-> ( 1 A 0 ) = ( M ` 1 ) ) )
253	252	rspcv 3145	. . . . . . . 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 65	. . . . . . 7 |- ( ph -> ( 1 A 0 ) = ( M ` 1 ) )
255		oveq2 6296	. . . . . . . . 9 |- ( s = 0 -> ( 1 A s ) = ( 1 A 0 ) )
256		eqid 2450	. . . . . . . . 9 |- ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) )
257		ovex 6316	. . . . . . . . 9 |- ( 1 A 0 ) e. _V
258	255, 256, 257	fvmpt 5946	. . . . . . . 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 5873	. . . . . . . . 9 |- ( M ` 1 ) e. _V
261	260	fvconst2 6118	. . . . . . . 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 2509	. . . . . 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 29999	. . . . 5 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) )
265		fconstmpt 4877	. . . . 5 |- ( ( 0 [,] 1 ) X. { ( M ` 1 ) } ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) )
266	264, 265	syl6eq 2500	. . . 4 |- ( ph -> ( s e. ( 0 [,] 1 ) |-> ( 1 A s ) ) = ( s e. ( 0 [,] 1 ) |-> ( M ` 1 ) ) )
267		mpteqb 5962	. . . . 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 6316	. . . . . 6 |- ( 1 A s ) e. _V
269	268	a1i 11	. . . . 5 |- ( s e. ( 0 [,] 1 ) -> ( 1 A s ) e. _V )
270	267, 269	mprg 2750	. . . 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 200	. . 3 |- ( ph -> A. s e. ( 0 [,] 1 ) ( 1 A s ) = ( M ` 1 ) )
272	271	r19.21bi 2756	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 A s ) = ( M ` 1 ) )
273	8, 16, 17, 91, 212, 213, 272	isphtpy2d 22011	1 |- ( ph -> A e. ( M ( PHtpy ` C ) N ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   A.wral 2736   E!wreu 2738   _Vcvv 3044   {csn 3967   <.cop 3973   U.cuni 4197   |-> cmpt 4460   X. cxp 4831   o. ccom 4837   Fn wfn 5576   -->wf 5577   ` cfv 5581   iota_crio 6249  (class class class)co 6288   0cc0 9536   1c1 9537   [,]cicc 11635   Topctop 19910  TopOnctopon 19911   Cn ccn 20233   Conccon 20419  𝑛Locally cnlly 20473   tX ctx 20568   IIcii 21900   Htpy chtpy 21991   PHtpycphtpy 21992   CovMap ccvm 29971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-ec 7362  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-cn 20236  df-cnp 20237  df-cmp 20395  df-con 20420  df-lly 20474  df-nlly 20475  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330  df-ii 21902  df-htpy 21994  df-phtpy 21995  df-phtpc 22016  df-pcon 29937  df-scon 29938  df-cvm 29972

This theorem is referenced by:  cvmliftpht  30034