Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmliftlem2 Unicode version

Theorem cvmliftlem2 23832
Description: Lemma for cvmlift 23845. 
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 8854 . . . 4  |-  0  e.  RR
32a1i 10 . . 3  |-  ( (
ph  /\  ps )  ->  0  e.  RR )
4 1re 8853 . . . 4  |-  1  e.  RR
54a1i 10 . . 3  |-  ( (
ph  /\  ps )  ->  1  e.  RR )
6 cvmliftlem1.m . . . . . . 7  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
7 elfznn 10835 . . . . . . 7  |-  ( M  e.  ( 1 ... N )  ->  M  e.  NN )
86, 7syl 15 . . . . . 6  |-  ( (
ph  /\  ps )  ->  M  e.  NN )
98nnred 9777 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  e.  RR )
10 peano2rem 9129 . . . . 5  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
119, 10syl 15 . . . 4  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  RR )
12 nnm1nn0 10021 . . . . . 6  |-  ( M  e.  NN  ->  ( M  -  1 )  e.  NN0 )
138, 12syl 15 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  NN0 )
1413nn0ge0d 10037 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <_  ( M  -  1 ) )
15 cvmliftlem.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
1615adantr 451 . . . . 5  |-  ( (
ph  /\  ps )  ->  N  e.  NN )
1716nnred 9777 . . . 4  |-  ( (
ph  /\  ps )  ->  N  e.  RR )
1816nngt0d 9805 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <  N )
19 divge0 9641 . . . 4  |-  ( ( ( ( M  - 
1 )  e.  RR  /\  0  <_  ( M  -  1 ) )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( ( M  -  1 )  /  N ) )
2011, 14, 17, 18, 19syl22anc 1183 . . 3  |-  ( (
ph  /\  ps )  ->  0  <_  ( ( M  -  1 )  /  N ) )
21 elfzle2 10816 . . . . . 6  |-  ( M  e.  ( 1 ... N )  ->  M  <_  N )
226, 21syl 15 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  <_  N )
2316nncnd 9778 . . . . . 6  |-  ( (
ph  /\  ps )  ->  N  e.  CC )
2423mulid1d 8868 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( N  x.  1 )  =  N )
2522, 24breqtrrd 4065 . . . 4  |-  ( (
ph  /\  ps )  ->  M  <_  ( N  x.  1 ) )
26 ledivmul 9645 . . . . 5  |-  ( ( M  e.  RR  /\  1  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
279, 5, 17, 18, 26syl112anc 1186 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
2825, 27mpbird 223 . . 3  |-  ( (
ph  /\  ps )  ->  ( M  /  N
)  <_  1 )
29 iccss 10734 . . 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 1183 . 2  |-  ( (
ph  /\  ps )  ->  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) ) 
C_  ( 0 [,] 1 ) )
311, 30syl5eqss 3235 1  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   U.cuni 3843   U_ciun 3921   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   (,)cioo 10672   [,]cicc 10675   ...cfz 10798   ↾t crest 13341   topGenctg 13358    Cn ccn 16970    Homeo chmeo 17460   IIcii 18395   CovMap ccvm 23801
This theorem is referenced by:  cvmliftlem3  23833  cvmliftlem6  23836  cvmliftlem8  23838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-icc 10679  df-fz 10799
  Copyright terms: Public domain W3C validator