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Theorem cvmliftlem2 27174
Description: Lemma for cvmlift 27187. 
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 9385 . . . 4  |-  0  e.  RR
32a1i 11 . . 3  |-  ( (
ph  /\  ps )  ->  0  e.  RR )
4 1re 9384 . . . 4  |-  1  e.  RR
54a1i 11 . . 3  |-  ( (
ph  /\  ps )  ->  1  e.  RR )
6 cvmliftlem1.m . . . . . . 7  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
7 elfznn 11477 . . . . . . 7  |-  ( M  e.  ( 1 ... N )  ->  M  e.  NN )
86, 7syl 16 . . . . . 6  |-  ( (
ph  /\  ps )  ->  M  e.  NN )
98nnred 10336 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  e.  RR )
10 peano2rem 9674 . . . . 5  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
119, 10syl 16 . . . 4  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  RR )
12 nnm1nn0 10620 . . . . . 6  |-  ( M  e.  NN  ->  ( M  -  1 )  e.  NN0 )
138, 12syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  NN0 )
1413nn0ge0d 10638 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <_  ( M  -  1 ) )
15 cvmliftlem.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
1615adantr 465 . . . . 5  |-  ( (
ph  /\  ps )  ->  N  e.  NN )
1716nnred 10336 . . . 4  |-  ( (
ph  /\  ps )  ->  N  e.  RR )
1816nngt0d 10364 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <  N )
19 divge0 10197 . . . 4  |-  ( ( ( ( M  - 
1 )  e.  RR  /\  0  <_  ( M  -  1 ) )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( ( M  -  1 )  /  N ) )
2011, 14, 17, 18, 19syl22anc 1219 . . 3  |-  ( (
ph  /\  ps )  ->  0  <_  ( ( M  -  1 )  /  N ) )
21 elfzle2 11454 . . . . . 6  |-  ( M  e.  ( 1 ... N )  ->  M  <_  N )
226, 21syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  <_  N )
2316nncnd 10337 . . . . . 6  |-  ( (
ph  /\  ps )  ->  N  e.  CC )
2423mulid1d 9402 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( N  x.  1 )  =  N )
2522, 24breqtrrd 4317 . . . 4  |-  ( (
ph  /\  ps )  ->  M  <_  ( N  x.  1 ) )
26 ledivmul 10204 . . . . 5  |-  ( ( M  e.  RR  /\  1  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
279, 5, 17, 18, 26syl112anc 1222 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
2825, 27mpbird 232 . . 3  |-  ( (
ph  /\  ps )  ->  ( M  /  N
)  <_  1 )
29 iccss 11362 . . 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 1219 . 2  |-  ( (
ph  /\  ps )  ->  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) ) 
C_  ( 0 [,] 1 ) )
311, 30syl5eqss 3399 1  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2714   {crab 2718    \ cdif 3324    i^i cin 3326    C_ wss 3327   (/)c0 3636   ~Pcpw 3859   {csn 3876   U.cuni 4090   U_ciun 4170   class class class wbr 4291    e. cmpt 4349    X. cxp 4837   `'ccnv 4838   ran crn 4840    |` cres 4841   "cima 4842   -->wf 5413   ` cfv 5417  (class class class)co 6090   1stc1st 6574   RRcr 9280   0cc0 9281   1c1 9282    x. cmul 9286    < clt 9417    <_ cle 9418    - cmin 9594    / cdiv 9992   NNcn 10321   NN0cn0 10578   (,)cioo 11299   [,]cicc 11302   ...cfz 11436   ↾t crest 14358   topGenctg 14375    Cn ccn 18827   Homeochmeo 19325   IIcii 20450   CovMap ccvm 27143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-icc 11306  df-fz 11437
This theorem is referenced by:  cvmliftlem3  27175  cvmliftlem6  27178  cvmliftlem8  27180
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