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Theorem cvmliftlem2 24978
Description: Lemma for cvmlift 24991. 
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 9096 . . . 4  |-  0  e.  RR
32a1i 11 . . 3  |-  ( (
ph  /\  ps )  ->  0  e.  RR )
4 1re 9095 . . . 4  |-  1  e.  RR
54a1i 11 . . 3  |-  ( (
ph  /\  ps )  ->  1  e.  RR )
6 cvmliftlem1.m . . . . . . 7  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
7 elfznn 11085 . . . . . . 7  |-  ( M  e.  ( 1 ... N )  ->  M  e.  NN )
86, 7syl 16 . . . . . 6  |-  ( (
ph  /\  ps )  ->  M  e.  NN )
98nnred 10020 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  e.  RR )
10 peano2rem 9372 . . . . 5  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
119, 10syl 16 . . . 4  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  RR )
12 nnm1nn0 10266 . . . . . 6  |-  ( M  e.  NN  ->  ( M  -  1 )  e.  NN0 )
138, 12syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  NN0 )
1413nn0ge0d 10282 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <_  ( M  -  1 ) )
15 cvmliftlem.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
1615adantr 453 . . . . 5  |-  ( (
ph  /\  ps )  ->  N  e.  NN )
1716nnred 10020 . . . 4  |-  ( (
ph  /\  ps )  ->  N  e.  RR )
1816nngt0d 10048 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <  N )
19 divge0 9884 . . . 4  |-  ( ( ( ( M  - 
1 )  e.  RR  /\  0  <_  ( M  -  1 ) )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( ( M  -  1 )  /  N ) )
2011, 14, 17, 18, 19syl22anc 1186 . . 3  |-  ( (
ph  /\  ps )  ->  0  <_  ( ( M  -  1 )  /  N ) )
21 elfzle2 11066 . . . . . 6  |-  ( M  e.  ( 1 ... N )  ->  M  <_  N )
226, 21syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  <_  N )
2316nncnd 10021 . . . . . 6  |-  ( (
ph  /\  ps )  ->  N  e.  CC )
2423mulid1d 9110 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( N  x.  1 )  =  N )
2522, 24breqtrrd 4241 . . . 4  |-  ( (
ph  /\  ps )  ->  M  <_  ( N  x.  1 ) )
26 ledivmul 9888 . . . . 5  |-  ( ( M  e.  RR  /\  1  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
279, 5, 17, 18, 26syl112anc 1189 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
2825, 27mpbird 225 . . 3  |-  ( (
ph  /\  ps )  ->  ( M  /  N
)  <_  1 )
29 iccss 10983 . . 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 1186 . 2  |-  ( (
ph  /\  ps )  ->  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) ) 
C_  ( 0 [,] 1 ) )
311, 30syl5eqss 3394 1  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017   U_ciun 4095   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   `'ccnv 4880   ran crn 4882    |` cres 4883   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084   1stc1st 6350   RRcr 8994   0cc0 8995   1c1 8996    x. cmul 9000    < clt 9125    <_ cle 9126    - cmin 9296    / cdiv 9682   NNcn 10005   NN0cn0 10226   (,)cioo 10921   [,]cicc 10924   ...cfz 11048   ↾t crest 13653   topGenctg 13670    Cn ccn 17293    Homeo chmeo 17790   IIcii 18910   CovMap ccvm 24947
This theorem is referenced by:  cvmliftlem3  24979  cvmliftlem6  24982  cvmliftlem8  24984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-icc 10928  df-fz 11049
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