Theorem cvmliftlem1 31267

Description: Lemma for cvmlift 31281. In cvmliftlem15 31280, we picked an N large enough so that the sections ( G " [ ( k - 1 ) / N , k / N ] ) are all contained in an even covering, and the function T enumerates these even coverings. So 1st ` ( T ` M ) is a neighborhood of ( G " [ ( M - 1 ) / N , M / N ] ), and 2nd ` ( T ` M ) is an even covering of 1st ` ( T ` M ), which is to say a disjoint union of open sets in C whose image is 1st ` ( T ` M ). (Contributed by Mario Carneiro, 14-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 (,) )
cvmliftlem1.m	|- ( ( ph /\ ps ) -> M e. ( 1 ... N ) )

Assertion

Ref	Expression
cvmliftlem1	|- ( ( ph /\ ps ) -> ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) )

Distinct variable groups:   v, B   j, k, s, u, v, F   j, M, k, s, u, v   P, k, u, v   C, j, k, s, u, v   ph, j, s   k, N, u, v   S, j, k, s, u, v   j, X   j, G, k, s, u, v   T, j, k, s, u, v   j, J, k, s, u, v

Allowed substitution hints:   ph( v, u, k)   ps( v, u, j, k, s)   B( u, j, k, s)   P( j, s)   L( v, u, j, k, s)   N( j, s)   X( v, u, k, s)

Proof of Theorem cvmliftlem1

Step	Hyp	Ref	Expression
1		relxp 5227	. . . . . 6 |- Rel ( { j } X. ( S ` j ) )
2	1	rgenw 2924	. . . . 5 |- A. j e. J Rel ( { j } X. ( S ` j ) )
3		reliun 5239	. . . . 5 |- ( Rel U_ j e. J ( { j } X. ( S ` j ) ) <-> A. j e. J Rel ( { j } X. ( S ` j ) ) )
4	2, 3	mpbir 221	. . . 4 |- Rel U_ j e. J ( { j } X. ( S ` j ) )
5		cvmliftlem.t	. . . . . 6 |- ( ph -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) )
6	5	adantr 481	. . . . 5 |- ( ( ph /\ ps ) -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) )
7		cvmliftlem1.m	. . . . 5 |- ( ( ph /\ ps ) -> M e. ( 1 ... N ) )
8	6, 7	ffvelrnd 6360	. . . 4 |- ( ( ph /\ ps ) -> ( T ` M ) e. U_ j e. J ( { j } X. ( S ` j ) ) )
9		1st2nd 7214	. . . 4 |- ( ( Rel U_ j e. J ( { j } X. ( S ` j ) ) /\ ( T ` M ) e. U_ j e. J ( { j } X. ( S ` j ) ) ) -> ( T ` M ) = <. ( 1st ` ( T ` M ) ) , ( 2nd ` ( T ` M ) ) >. )
10	4, 8, 9	sylancr 695	. . 3 |- ( ( ph /\ ps ) -> ( T ` M ) = <. ( 1st ` ( T ` M ) ) , ( 2nd ` ( T ` M ) ) >. )
11	10, 8	eqeltrrd 2702	. 2 |- ( ( ph /\ ps ) -> <. ( 1st ` ( T ` M ) ) , ( 2nd ` ( T ` M ) ) >. e. U_ j e. J ( { j } X. ( S ` j ) ) )
12		fveq2 6191	. . . 4 |- ( j = ( 1st ` ( T ` M ) ) -> ( S ` j ) = ( S ` ( 1st ` ( T ` M ) ) ) )
13	12	opeliunxp2 5260	. . 3 |- ( <. ( 1st ` ( T ` M ) ) , ( 2nd ` ( T ` M ) ) >. e. U_ j e. J ( { j } X. ( S ` j ) ) <-> ( ( 1st ` ( T ` M ) ) e. J /\ ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) ) )
14	13	simprbi 480	. 2 |- ( <. ( 1st ` ( T ` M ) ) , ( 2nd ` ( T ` M ) ) >. e. U_ j e. J ( { j } X. ( S ` j ) ) -> ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) )
15	11, 14	syl 17	1 |- ( ( ph /\ ps ) -> ( 2nd ` ( T ` M ) ) e. ( S ` ( 1st ` ( T ` M ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   1c1 9937   - cmin 10266   / cdiv 10684   NNcn 11020   (,)cioo 12175   [,]cicc 12178   ...cfz 12326   ↾_t crest 16081   topGenctg 16098   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  cvmliftlem6  31272  cvmliftlem8  31274  cvmliftlem9  31275