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Theorem cvmliftlem2 28928
Description: Lemma for cvmlift 28941. 
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 0red 9614 . . 3  |-  ( (
ph  /\  ps )  ->  0  e.  RR )
3 1red 9628 . . 3  |-  ( (
ph  /\  ps )  ->  1  e.  RR )
4 cvmliftlem1.m . . . . . . 7  |-  ( (
ph  /\  ps )  ->  M  e.  ( 1 ... N ) )
5 elfznn 11739 . . . . . . 7  |-  ( M  e.  ( 1 ... N )  ->  M  e.  NN )
64, 5syl 16 . . . . . 6  |-  ( (
ph  /\  ps )  ->  M  e.  NN )
76nnred 10571 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  e.  RR )
8 peano2rem 9905 . . . . 5  |-  ( M  e.  RR  ->  ( M  -  1 )  e.  RR )
97, 8syl 16 . . . 4  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  RR )
10 nnm1nn0 10858 . . . . . 6  |-  ( M  e.  NN  ->  ( M  -  1 )  e.  NN0 )
116, 10syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( M  -  1 )  e.  NN0 )
1211nn0ge0d 10876 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <_  ( M  -  1 ) )
13 cvmliftlem.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
1413adantr 465 . . . . 5  |-  ( (
ph  /\  ps )  ->  N  e.  NN )
1514nnred 10571 . . . 4  |-  ( (
ph  /\  ps )  ->  N  e.  RR )
1614nngt0d 10600 . . . 4  |-  ( (
ph  /\  ps )  ->  0  <  N )
17 divge0 10432 . . . 4  |-  ( ( ( ( M  - 
1 )  e.  RR  /\  0  <_  ( M  -  1 ) )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( ( M  -  1 )  /  N ) )
189, 12, 15, 16, 17syl22anc 1229 . . 3  |-  ( (
ph  /\  ps )  ->  0  <_  ( ( M  -  1 )  /  N ) )
19 elfzle2 11715 . . . . . 6  |-  ( M  e.  ( 1 ... N )  ->  M  <_  N )
204, 19syl 16 . . . . 5  |-  ( (
ph  /\  ps )  ->  M  <_  N )
2114nncnd 10572 . . . . . 6  |-  ( (
ph  /\  ps )  ->  N  e.  CC )
2221mulid1d 9630 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( N  x.  1 )  =  N )
2320, 22breqtrrd 4482 . . . 4  |-  ( (
ph  /\  ps )  ->  M  <_  ( N  x.  1 ) )
24 ledivmul 10439 . . . . 5  |-  ( ( M  e.  RR  /\  1  e.  RR  /\  ( N  e.  RR  /\  0  <  N ) )  -> 
( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
257, 3, 15, 16, 24syl112anc 1232 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ( M  /  N )  <_  1  <->  M  <_  ( N  x.  1 ) ) )
2623, 25mpbird 232 . . 3  |-  ( (
ph  /\  ps )  ->  ( M  /  N
)  <_  1 )
27 iccss 11617 . . 3  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
( ( M  - 
1 )  /  N
)  /\  ( M  /  N )  <_  1
) )  ->  (
( ( M  - 
1 )  /  N
) [,] ( M  /  N ) ) 
C_  ( 0 [,] 1 ) )
282, 3, 18, 26, 27syl22anc 1229 . 2  |-  ( (
ph  /\  ps )  ->  ( ( ( M  -  1 )  /  N ) [,] ( M  /  N ) ) 
C_  ( 0 [,] 1 ) )
291, 28syl5eqss 3543 1  |-  ( (
ph  /\  ps )  ->  W  C_  ( 0 [,] 1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   {csn 4032   U.cuni 4251   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   ran crn 5009    |` cres 5010   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   NN0cn0 10816   (,)cioo 11554   [,]cicc 11557   ...cfz 11697   ↾t crest 14838   topGenctg 14855    Cn ccn 19852   Homeochmeo 20380   IIcii 21505   CovMap ccvm 28897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-icc 11561  df-fz 11698
This theorem is referenced by:  cvmliftlem3  28929  cvmliftlem6  28932  cvmliftlem8  28934
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