Theorem cvmliftlem3 31269

Description: Lemma for cvmlift 31281. Since 1st ` ( T ` M ) is a neighborhood of ( G " W ), every element A e. W satisfies ( G ` A ) e. ( 1st ` ( T ` M ) ). (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 (,) )
cvmliftlem1.m	|- ( ( ph /\ ps ) -> M e. ( 1 ... N ) )
cvmliftlem3.3	|- W = ( ( ( M - 1 ) / N ) [,] ( M / N ) )
cvmliftlem3.m	|- ( ( ph /\ ps ) -> A e. W )

Assertion

Ref	Expression
cvmliftlem3	|- ( ( ph /\ ps ) -> ( G ` A ) e. ( 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   k, W

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

Proof of Theorem cvmliftlem3

Step	Hyp	Ref	Expression
1		cvmliftlem1.m	. . 3 |- ( ( ph /\ ps ) -> M e. ( 1 ... N ) )
2		cvmliftlem.a	. . . 4 |- ( ph -> A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) )
3	2	adantr 481	. . 3 |- ( ( ph /\ ps ) -> A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) )
4		oveq1 6657	. . . . . . . . 9 |- ( k = M -> ( k - 1 ) = ( M - 1 ) )
5	4	oveq1d 6665	. . . . . . . 8 |- ( k = M -> ( ( k - 1 ) / N ) = ( ( M - 1 ) / N ) )
6		oveq1 6657	. . . . . . . 8 |- ( k = M -> ( k / N ) = ( M / N ) )
7	5, 6	oveq12d 6668	. . . . . . 7 |- ( k = M -> ( ( ( k - 1 ) / N ) [,] ( k / N ) ) = ( ( ( M - 1 ) / N ) [,] ( M / N ) ) )
8		cvmliftlem3.3	. . . . . . 7 |- W = ( ( ( M - 1 ) / N ) [,] ( M / N ) )
9	7, 8	syl6eqr 2674	. . . . . 6 |- ( k = M -> ( ( ( k - 1 ) / N ) [,] ( k / N ) ) = W )
10	9	imaeq2d 5466	. . . . 5 |- ( k = M -> ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) = ( G " W ) )
11		fveq2 6191	. . . . . 6 |- ( k = M -> ( T ` k ) = ( T ` M ) )
12	11	fveq2d 6195	. . . . 5 |- ( k = M -> ( 1st ` ( T ` k ) ) = ( 1st ` ( T ` M ) ) )
13	10, 12	sseq12d 3634	. . . 4 |- ( k = M -> ( ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) <-> ( G " W ) C_ ( 1st ` ( T ` M ) ) ) )
14	13	rspcv 3305	. . 3 |- ( M e. ( 1 ... N ) -> ( A. k e. ( 1 ... N ) ( G " ( ( ( k - 1 ) / N ) [,] ( k / N ) ) ) C_ ( 1st ` ( T ` k ) ) -> ( G " W ) C_ ( 1st ` ( T ` M ) ) ) )
15	1, 3, 14	sylc 65	. 2 |- ( ( ph /\ ps ) -> ( G " W ) C_ ( 1st ` ( T ` M ) ) )
16		cvmliftlem3.m	. . 3 |- ( ( ph /\ ps ) -> A e. W )
17		cvmliftlem.g	. . . . . . 7 |- ( ph -> G e. ( II Cn J ) )
18		iiuni 22684	. . . . . . . 8 |- ( 0 [,] 1 ) = U. II
19		cvmliftlem.x	. . . . . . . 8 |- X = U. J
20	18, 19	cnf 21050	. . . . . . 7 |- ( G e. ( II Cn J ) -> G : ( 0 [,] 1 ) --> X )
21	17, 20	syl 17	. . . . . 6 |- ( ph -> G : ( 0 [,] 1 ) --> X )
22	21	adantr 481	. . . . 5 |- ( ( ph /\ ps ) -> G : ( 0 [,] 1 ) --> X )
23		ffun 6048	. . . . 5 |- ( G : ( 0 [,] 1 ) --> X -> Fun G )
24	22, 23	syl 17	. . . 4 |- ( ( ph /\ ps ) -> Fun G )
25		cvmliftlem.1	. . . . . 6 |- 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 ) ) ) ) } )
26		cvmliftlem.b	. . . . . 6 |- B = U. C
27		cvmliftlem.f	. . . . . 6 |- ( ph -> F e. ( C CovMap J ) )
28		cvmliftlem.p	. . . . . 6 |- ( ph -> P e. B )
29		cvmliftlem.e	. . . . . 6 |- ( ph -> ( F ` P ) = ( G ` 0 ) )
30		cvmliftlem.n	. . . . . 6 |- ( ph -> N e. NN )
31		cvmliftlem.t	. . . . . 6 |- ( ph -> T : ( 1 ... N ) --> U_ j e. J ( { j } X. ( S ` j ) ) )
32		cvmliftlem.l	. . . . . 6 |- L = ( topGen ` ran (,) )
33	25, 26, 19, 27, 17, 28, 29, 30, 31, 2, 32, 1, 8	cvmliftlem2 31268	. . . . 5 |- ( ( ph /\ ps ) -> W C_ ( 0 [,] 1 ) )
34		fdm 6051	. . . . . 6 |- ( G : ( 0 [,] 1 ) --> X -> dom G = ( 0 [,] 1 ) )
35	22, 34	syl 17	. . . . 5 |- ( ( ph /\ ps ) -> dom G = ( 0 [,] 1 ) )
36	33, 35	sseqtr4d 3642	. . . 4 |- ( ( ph /\ ps ) -> W C_ dom G )
37		funfvima2 6493	. . . 4 |- ( ( Fun G /\ W C_ dom G ) -> ( A e. W -> ( G ` A ) e. ( G " W ) ) )
38	24, 36, 37	syl2anc 693	. . 3 |- ( ( ph /\ ps ) -> ( A e. W -> ( G ` A ) e. ( G " W ) ) )
39	16, 38	mpd 15	. 2 |- ( ( ph /\ ps ) -> ( G ` A ) e. ( G " W ) )
40	15, 39	sseldd 3604	1 |- ( ( ph /\ ps ) -> ( G ` A ) e. ( 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   U.cuni 4436   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   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  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-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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  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-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-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-ii 22680

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