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Theorem cvmliftlem2 24926
Description: Lemma for cvmlift 24939. 
W  =  [ ( k  -  1 )  /  N ,  k  /  N ] is a subset of  [ 0 ,  1 ] for each  M  e.  ( 1 ... N
). (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.  ( ( Ct  u ) 
Homeo  ( Jt  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 ) )
Assertion
Ref Expression
cvmliftlem2  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
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)    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 cvmliftlem2
StepHypRef Expression
1 cvmliftlem3.3 . 2  |-  W  =  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) )
2 0re 9047 . . . 4  |-  0  e.  RR
32a1i 11 . . 3  |-  ( (
ph  /\  ps )  ->  0  e.  RR )
4 1re 9046 . . . 4  |-  1  e.  RR
54a1i 11 . . 3  |-  ( (
ph  /\  ps )  ->  1  e.  RR )
6 cvmliftlem1.m . . . . . . 7  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
7 elfznn 11036 . . . . . . 7  |-  ( M  e.  ( 1 ... N )  ->  M  e.  NN )
86, 7syl 16 . . . . . 6  |-  ( (
ph  /\  ps )  ->  M  e.  NN )
98nnred 9971 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  e.  RR )
10 peano2rem 9323 . . . . 5  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
119, 10syl 16 . . . 4  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  RR )
12 nnm1nn0 10217 . . . . . 6  |-  ( M  e.  NN  ->  ( M  -  1 )  e.  NN0 )
138, 12syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  NN0 )
1413nn0ge0d 10233 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <_  ( M  -  1 ) )
15 cvmliftlem.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
1615adantr 452 . . . . 5  |-  ( (
ph  /\  ps )  ->  N  e.  NN )
1716nnred 9971 . . . 4  |-  ( (
ph  /\  ps )  ->  N  e.  RR )
1816nngt0d 9999 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <  N )
19 divge0 9835 . . . 4  |-  ( ( ( ( M  - 
1 )  e.  RR  /\  0  <_  ( M  -  1 ) )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( ( M  -  1 )  /  N ) )
2011, 14, 17, 18, 19syl22anc 1185 . . 3  |-  ( (
ph  /\  ps )  ->  0  <_  ( ( M  -  1 )  /  N ) )
21 elfzle2 11017 . . . . . 6  |-  ( M  e.  ( 1 ... N )  ->  M  <_  N )
226, 21syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  <_  N )
2316nncnd 9972 . . . . . 6  |-  ( (
ph  /\  ps )  ->  N  e.  CC )
2423mulid1d 9061 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( N  x.  1 )  =  N )
2522, 24breqtrrd 4198 . . . 4  |-  ( (
ph  /\  ps )  ->  M  <_  ( N  x.  1 ) )
26 ledivmul 9839 . . . . 5  |-  ( ( M  e.  RR  /\  1  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
279, 5, 17, 18, 26syl112anc 1188 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
2825, 27mpbird 224 . . 3  |-  ( (
ph  /\  ps )  ->  ( M  /  N
)  <_  1 )
29 iccss 10934 . . 3  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
( ( M  - 
1 )  /  N
)  /\  ( M  /  N )  <_  1
) )  ->  (
( ( M  - 
1 )  /  N
) [,] ( M  /  N ) ) 
C_  ( 0 [,] 1 ) )
303, 5, 20, 28, 29syl22anc 1185 . 2  |-  ( (
ph  /\  ps )  ->  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) ) 
C_  ( 0 [,] 1 ) )
311, 30syl5eqss 3352 1  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   U.cuni 3975   U_ciun 4053   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   `'ccnv 4836   ran crn 4838    |` cres 4839   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   (,)cioo 10872   [,]cicc 10875   ...cfz 10999   ↾t crest 13603   topGenctg 13620    Cn ccn 17242    Homeo chmeo 17738   IIcii 18858   CovMap ccvm 24895
This theorem is referenced by:  cvmliftlem3  24927  cvmliftlem6  24930  cvmliftlem8  24932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-icc 10879  df-fz 11000
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