Theorem cvmliftlem8 31274

Description: Lemma for cvmlift 31281. The functions Q are continuous functions because they are defined as `' ( F |` I ) o. G where G is continuous and ( F |` I ) is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.)

Hypotheses

Ref	Expression
cvmliftlem.1	|- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
cvmliftlem.b	|- B = U. C
cvmliftlem.x	|- X = U. J
cvmliftlem.f	|- ( ph -> F e. ( C CovMap J ) )
cvmliftlem.g	|- ( ph -> G e. ( II Cn J ) )
cvmliftlem.p	|- ( ph -> P e. B )
cvmliftlem.e	|- ( ph -> ( F ` P ) = ( G ` 0 ) )
cvmliftlem.n	|- ( ph -> N e. NN )
cvmliftlem.t	|- ( ph -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) )
cvmliftlem.a	|- ( ph -> A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) )
cvmliftlem.l	|- L = ( topGen ` ran (,) )
cvmliftlem.q	|- Q = seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) )
cvmliftlem5.3	|- W = ( ( ( M - 1 ) / N ) [,] ( M / N ) )

Assertion

Ref	Expression
cvmliftlem8	|- ( ( ph /\ M e. ( 1 ... N ) ) -> ( Q ` M ) e. ( ( L ↾t W ) Cn C ) )

Distinct variable groups:   v, b, z, B   j, b, k, m, s, u, x, F, v, z   z, L   M, b, j, k, m, s, u, v, x, z   P, b, k, m, u, v, x, z   C, b, j, k, s, u, v, z   ph, j, s, x, z   N, b, k, m, u, v, x, z   S, b, j, k, s, u, v, x, z   j, X   G, b, j, k, m, s, u, v, x, z   T, b, j, k, m, s, u, v, x, z   J, b, j, k, s, u, v, x, z   Q, b, k, m, u, v, x, z   k, W, m, x, z

Allowed substitution hints:   ph( v, u, k, m, b)   B( x, u, j, k, m, s)   C( x, m)   P( j, s)   Q( j, s)   S( m)   J( m)   L( x, v, u, j, k, m, s, b)   N( j, s)   W( v, u, j, s, b)   X( x, z, v, u, k, m, s, b)

Proof of Theorem cvmliftlem8

Step	Hyp	Ref	Expression
1		elfznn 12370	. . 3 |- ( M e. ( 1 ... N ) -> M e. NN )
2		cvmliftlem.1	. . . 4 |- S = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. u e. s ( A. v e. ( s \ { u } ) ( u i^i v ) = (/) /\ ( F |` u ) e. ( ( C ↾t u ) Homeo ( J ↾t k ) ) ) ) } )
3		cvmliftlem.b	. . . 4 |- B = U. C
4		cvmliftlem.x	. . . 4 |- X = U. J
5		cvmliftlem.f	. . . 4 |- ( ph -> F e. ( C CovMap J ) )
6		cvmliftlem.g	. . . 4 |- ( ph -> G e. ( II Cn J ) )
7		cvmliftlem.p	. . . 4 |- ( ph -> P e. B )
8		cvmliftlem.e	. . . 4 |- ( ph -> ( F ` P ) = ( G ` 0 ) )
9		cvmliftlem.n	. . . 4 |- ( ph -> N e. NN )
10		cvmliftlem.t	. . . 4 |- ( ph -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) )
11		cvmliftlem.a	. . . 4 |- ( ph -> A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) )
12		cvmliftlem.l	. . . 4 |- L = ( topGen ` ran (,) )
13		cvmliftlem.q	. . . 4 |- Q = seq 0 ( ( x e. _V , m e. NN |-> ( z e. ( ( ( m - 1 ) / N ) [,] ( m / N ) ) |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` m ) ) ( x ` ( ( m - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) ) , ( ( _I |` NN ) u. { <. 0 , { <. 0 , P >. } >. } ) )
14		cvmliftlem5.3	. . . 4 |- W = ( ( ( M - 1 ) / N ) [,] ( M / N ) )
15	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cvmliftlem5 31271	. . 3 |- ( ( ph /\ M e. NN ) -> ( Q ` M ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) )
16	1, 15	sylan2 491	. 2 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( Q ` M ) = ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) )
17	5	adantr 481	. . . 4 |- ( ( ph /\ M e. ( 1 ... N ) ) -> F e. ( C CovMap J ) )
18		cvmtop1 31242	. . . 4 |- ( F e. ( C CovMap J ) -> C e. Top )
19		cnrest2r 21091	. . . 4 |- ( C e. Top -> ( ( L ↾t W ) Cn ( C ↾t ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ) C_ ( ( L ↾t W ) Cn C ) )
20	17, 18, 19	3syl 18	. . 3 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( L ↾t W ) Cn ( C ↾t ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ) C_ ( ( L ↾t W ) Cn C ) )
21		retopon 22567	. . . . . 6 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
22	12, 21	eqeltri 2697	. . . . 5 |- L e. (TopOn ` RR )
23		simpr 477	. . . . . . 7 |- ( ( ph /\ M e. ( 1 ... N ) ) -> M e. ( 1 ... N ) )
24	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14	cvmliftlem2 31268	. . . . . 6 |- ( ( ph /\ M e. ( 1 ... N ) ) -> W C_ ( 0 [,] 1 ) )
25		unitssre 12319	. . . . . 6 |- ( 0 [,] 1 ) C_ RR
26	24, 25	syl6ss 3615	. . . . 5 |- ( ( ph /\ M e. ( 1 ... N ) ) -> W C_ RR )
27		resttopon 20965	. . . . 5 |- ( ( L e. (TopOn ` RR ) /\ W C_ RR ) -> ( L ↾t W ) e. (TopOn ` W ) )
28	22, 26, 27	sylancr 695	. . . 4 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( L ↾t W ) e. (TopOn ` W ) )
29		eqid 2622	. . . . . . 7 |- ( II ↾t W ) = ( II ↾t W )
30		iitopon 22682	. . . . . . . 8 |- II e. (TopOn ` ( 0 [,] 1 ) )
31	30	a1i 11	. . . . . . 7 |- ( ( ph /\ M e. ( 1 ... N ) ) -> II e. (TopOn ` ( 0 [,] 1 ) ) )
32	6	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ M e. ( 1 ... N ) ) -> G e. ( II Cn J ) )
33		iiuni 22684	. . . . . . . . . . 11 |- ( 0 [,] 1 ) = U. II
34	33, 4	cnf 21050	. . . . . . . . . 10 |- ( G e. ( II Cn J ) -> G : ( 0 [,] 1 ) --> X )
35	32, 34	syl 17	. . . . . . . . 9 |- ( ( ph /\ M e. ( 1 ... N ) ) -> G : ( 0 [,] 1 ) --> X )
36	35	feqmptd 6249	. . . . . . . 8 |- ( ( ph /\ M e. ( 1 ... N ) ) -> G = ( z e. ( 0 [,] 1 ) |-> ( G ` z ) ) )
37	36, 32	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( z e. ( 0 [,] 1 ) |-> ( G ` z ) ) e. ( II Cn J ) )
38	29, 31, 24, 37	cnmpt1res 21479	. . . . . 6 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( z e. W |-> ( G ` z ) ) e. ( ( II ↾t W ) Cn J ) )
39		dfii2 22685	. . . . . . . . . 10 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
40	12	oveq1i 6660	. . . . . . . . . 10 |- ( L ↾t ( 0 [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
41	39, 40	eqtr4i 2647	. . . . . . . . 9 |- II = ( L ↾t ( 0 [,] 1 ) )
42	41	oveq1i 6660	. . . . . . . 8 |- ( II ↾t W ) = ( ( L ↾t ( 0 [,] 1 ) ) ↾t W )
43		retop 22565	. . . . . . . . . . 11 |- ( topGen ` ran (,) ) e. Top
44	12, 43	eqeltri 2697	. . . . . . . . . 10 |- L e. Top
45	44	a1i 11	. . . . . . . . 9 |- ( ( ph /\ M e. ( 1 ... N ) ) -> L e. Top )
46		ovexd 6680	. . . . . . . . 9 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( 0 [,] 1 ) e. _V )
47		restabs 20969	. . . . . . . . 9 |- ( ( L e. Top /\ W C_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) e. _V ) -> ( ( L ↾t ( 0 [,] 1 ) ) ↾t W ) = ( L ↾t W ) )
48	45, 24, 46, 47	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( L ↾t ( 0 [,] 1 ) ) ↾t W ) = ( L ↾t W ) )
49	42, 48	syl5eq 2668	. . . . . . 7 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( II ↾t W ) = ( L ↾t W ) )
50	49	oveq1d 6665	. . . . . 6 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( II ↾t W ) Cn J ) = ( ( L ↾t W ) Cn J ) )
51	38, 50	eleqtrd 2703	. . . . 5 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( z e. W |-> ( G ` z ) ) e. ( ( L ↾t W ) Cn J ) )
52		cvmtop2 31243	. . . . . . . 8 |- ( F e. ( C CovMap J ) -> J e. Top )
53	17, 52	syl 17	. . . . . . 7 |- ( ( ph /\ M e. ( 1 ... N ) ) -> J e. Top )
54	4	toptopon 20722	. . . . . . 7 |- ( J e. Top <-> J e. (TopOn ` X ) )
55	53, 54	sylib 208	. . . . . 6 |- ( ( ph /\ M e. ( 1 ... N ) ) -> J e. (TopOn ` X ) )
56		simprl 794	. . . . . . . . . 10 |- ( ( ph /\ ( M e. ( 1 ... N ) /\ z e. W ) ) -> M e. ( 1 ... N ) )
57		simprr 796	. . . . . . . . . 10 |- ( ( ph /\ ( M e. ( 1 ... N ) /\ z e. W ) ) -> z e. W )
58	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 56, 14, 57	cvmliftlem3 31269	. . . . . . . . 9 |- ( ( ph /\ ( M e. ( 1 ... N ) /\ z e. W ) ) -> ( G ` z ) e. ( 1st ` ( T ` M ) ) )
59	58	anassrs 680	. . . . . . . 8 |- ( ( ( ph /\ M e. ( 1 ... N ) ) /\ z e. W ) -> ( G ` z ) e. ( 1st ` ( T ` M ) ) )
60		eqid 2622	. . . . . . . 8 |- ( z e. W |-> ( G ` z ) ) = ( z e. W |-> ( G ` z ) )
61	59, 60	fmptd 6385	. . . . . . 7 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( z e. W |-> ( G ` z ) ) : W --> ( 1st ` ( T ` M ) ) )
62		frn 6053	. . . . . . 7 |- ( ( z e. W |-> ( G ` z ) ) : W --> ( 1st ` ( T ` M ) ) -> ran ( z e. W |-> ( G ` z ) ) C_ ( 1st ` ( T ` M ) ) )
63	61, 62	syl 17	. . . . . 6 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ran ( z e. W |-> ( G ` z ) ) C_ ( 1st ` ( T ` M ) ) )
64	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23	cvmliftlem1 31267	. . . . . . . 8 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) )
65	2	cvmsrcl 31246	. . . . . . . 8 |- ( ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) -> ( 1st ` ( T ` M ) ) e. J )
66		elssuni 4467	. . . . . . . 8 |- ( ( 1st ` ( T ` M ) ) e. J -> ( 1st ` ( T ` M ) ) C_ U. J )
67	64, 65, 66	3syl 18	. . . . . . 7 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( 1st ` ( T ` M ) ) C_ U. J )
68	67, 4	syl6sseqr 3652	. . . . . 6 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( 1st ` ( T ` M ) ) C_ X )
69		cnrest2 21090	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ ran ( z e. W |-> ( G ` z ) ) C_ ( 1st ` ( T ` M ) ) /\ ( 1st ` ( T ` M ) ) C_ X ) -> ( ( z e. W |-> ( G ` z ) ) e. ( ( L ↾t W ) Cn J ) <-> ( z e. W |-> ( G ` z ) ) e. ( ( L ↾t W ) Cn ( J ↾t ( 1st ` ( T ` M ) ) ) ) ) )
70	55, 63, 68, 69	syl3anc 1326	. . . . 5 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( z e. W |-> ( G ` z ) ) e. ( ( L ↾t W ) Cn J ) <-> ( z e. W |-> ( G ` z ) ) e. ( ( L ↾t W ) Cn ( J ↾t ( 1st ` ( T ` M ) ) ) ) ) )
71	51, 70	mpbid 222	. . . 4 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( z e. W |-> ( G ` z ) ) e. ( ( L ↾t W ) Cn ( J ↾t ( 1st ` ( T ` M ) ) ) ) )
72	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cvmliftlem7 31273	. . . . . . . . . 10 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( `' F " { ( G ` ( ( M - 1 ) / N ) ) } ) )
73		cvmcn 31244	. . . . . . . . . . . 12 |- ( F e. ( C CovMap J ) -> F e. ( C Cn J ) )
74	3, 4	cnf 21050	. . . . . . . . . . . 12 |- ( F e. ( C Cn J ) -> F : B --> X )
75	17, 73, 74	3syl 18	. . . . . . . . . . 11 |- ( ( ph /\ M e. ( 1 ... N ) ) -> F : B --> X )
76		ffn 6045	. . . . . . . . . . 11 |- ( F : B --> X -> F Fn B )
77		fniniseg 6338	. . . . . . . . . . 11 |- ( F Fn B -> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( `' F " { ( G ` ( ( M - 1 ) / N ) ) } ) <-> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B /\ ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) = ( G ` ( ( M - 1 ) / N ) ) ) ) )
78	75, 76, 77	3syl 18	. . . . . . . . . 10 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( `' F " { ( G ` ( ( M - 1 ) / N ) ) } ) <-> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B /\ ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) = ( G ` ( ( M - 1 ) / N ) ) ) ) )
79	72, 78	mpbid 222	. . . . . . . . 9 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B /\ ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) = ( G ` ( ( M - 1 ) / N ) ) ) )
80	79	simpld 475	. . . . . . . 8 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B )
81	79	simprd 479	. . . . . . . . 9 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) = ( G ` ( ( M - 1 ) / N ) ) )
82	1	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ M e. ( 1 ... N ) ) -> M e. NN )
83	82	nnred 11035	. . . . . . . . . . . . . . 15 |- ( ( ph /\ M e. ( 1 ... N ) ) -> M e. RR )
84		peano2rem 10348	. . . . . . . . . . . . . . 15 |- ( M e. RR -> ( M - 1 ) e. RR )
85	83, 84	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( M - 1 ) e. RR )
86	9	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ M e. ( 1 ... N ) ) -> N e. NN )
87	85, 86	nndivred 11069	. . . . . . . . . . . . 13 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( M - 1 ) / N ) e. RR )
88	87	rexrd 10089	. . . . . . . . . . . 12 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( M - 1 ) / N ) e. RR* )
89	83, 86	nndivred 11069	. . . . . . . . . . . . 13 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( M / N ) e. RR )
90	89	rexrd 10089	. . . . . . . . . . . 12 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( M / N ) e. RR* )
91	83	ltm1d 10956	. . . . . . . . . . . . . 14 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( M - 1 ) < M )
92	86	nnred 11035	. . . . . . . . . . . . . . 15 |- ( ( ph /\ M e. ( 1 ... N ) ) -> N e. RR )
93	86	nngt0d 11064	. . . . . . . . . . . . . . 15 |- ( ( ph /\ M e. ( 1 ... N ) ) -> 0 < N )
94		ltdiv1 10887	. . . . . . . . . . . . . . 15 |- ( ( ( M - 1 ) e. RR /\ M e. RR /\ ( N e. RR /\ 0 < N ) ) -> ( ( M - 1 ) < M <-> ( ( M - 1 ) / N ) < ( M / N ) ) )
95	85, 83, 92, 93, 94	syl112anc 1330	. . . . . . . . . . . . . 14 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( M - 1 ) < M <-> ( ( M - 1 ) / N ) < ( M / N ) ) )
96	91, 95	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( M - 1 ) / N ) < ( M / N ) )
97	87, 89, 96	ltled 10185	. . . . . . . . . . . 12 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( M - 1 ) / N ) <_ ( M / N ) )
98		lbicc2 12288	. . . . . . . . . . . 12 |- ( ( ( ( M - 1 ) / N ) e. RR* /\ ( M / N ) e. RR* /\ ( ( M - 1 ) / N ) <_ ( M / N ) ) -> ( ( M - 1 ) / N ) e. ( ( ( M - 1 ) / N ) [,] ( M / N ) ) )
99	88, 90, 97, 98	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( M - 1 ) / N ) e. ( ( ( M - 1 ) / N ) [,] ( M / N ) ) )
100	99, 14	syl6eleqr 2712	. . . . . . . . . 10 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( M - 1 ) / N ) e. W )
101	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 100	cvmliftlem3 31269	. . . . . . . . 9 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( G ` ( ( M - 1 ) / N ) ) e. ( 1st ` ( T ` M ) ) )
102	81, 101	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) e. ( 1st ` ( T ` M ) ) )
103		eqid 2622	. . . . . . . . 9 |- ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) = ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b )
104	2, 3, 103	cvmsiota 31259	. . . . . . . 8 |- ( ( F e. ( C CovMap J ) /\ ( ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) /\ ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. B /\ ( F ` ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) ) e. ( 1st ` ( T ` M ) ) ) ) -> ( ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. ( 2nd ` ( T ` M ) ) /\ ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) )
105	17, 64, 80, 102, 104	syl13anc 1328	. . . . . . 7 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. ( 2nd ` ( T ` M ) ) /\ ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) )
106	105	simpld 475	. . . . . 6 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. ( 2nd ` ( T ` M ) ) )
107	2	cvmshmeo 31253	. . . . . 6 |- ( ( ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) /\ ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) e. ( 2nd ` ( T ` M ) ) ) -> ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) e. ( ( C ↾t ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) Homeo ( J ↾t ( 1st ` ( T ` M ) ) ) ) )
108	64, 106, 107	syl2anc 693	. . . . 5 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) e. ( ( C ↾t ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) Homeo ( J ↾t ( 1st ` ( T ` M ) ) ) ) )
109		hmeocnvcn 21564	. . . . 5 |- ( ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) e. ( ( C ↾t ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) Homeo ( J ↾t ( 1st ` ( T ` M ) ) ) ) -> `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) e. ( ( J ↾t ( 1st ` ( T ` M ) ) ) Cn ( C ↾t ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ) )
110	108, 109	syl 17	. . . 4 |- ( ( ph /\ M e. ( 1 ... N ) ) -> `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) e. ( ( J ↾t ( 1st ` ( T ` M ) ) ) Cn ( C ↾t ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ) )
111	28, 71, 110	cnmpt11f 21467	. . 3 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) e. ( ( L ↾t W ) Cn ( C ↾t ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ) )
112	20, 111	sseldd 3604	. 2 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( z e. W |-> ( `' ( F |` ( iota_ b e. ( 2nd ` ( T ` M ) ) ( ( Q ` ( M - 1 ) ) ` ( ( M - 1 ) / N ) ) e. b ) ) ` ( G ` z ) ) ) e. ( ( L ↾t W ) Cn C ) )
113	16, 112	eqeltrd 2701	1 |- ( ( ph /\ M e. ( 1 ... N ) ) -> ( Q ` M ) e. ( ( L ↾t W ) Cn C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   (,)cioo 12175   [,]cicc 12178   ...cfz 12326   seqcseq 12801   ↾_t crest 16081   topGenctg 16098   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   Homeochmeo 21556   IIcii 22678   CovMap ccvm 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-hmeo 21558  df-ii 22680  df-cvm 31238

This theorem is referenced by:  cvmliftlem10  31276