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Theorem cvmliftlem13 24983
Description: Lemma for cvmlift 24986. The initial value of  K is  P because  Q ( 1 ) is a subset of  K which takes value  P at  0. (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  (,) )
cvmliftlem.q  |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N
) )  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m
) ) ( x `
 ( ( m  -  1 )  /  N ) )  e.  b ) ) `  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) )
cvmliftlem.k  |-  K  = 
U_ k  e.  ( 1 ... N ) ( Q `  k
)
Assertion
Ref Expression
cvmliftlem13  |-  ( ph  ->  ( K `  0
)  =  P )
Distinct variable groups:    v, b,
z, B    j, b,
k, m, s, u, x, F, v, z   
z, L    P, b,
k, m, u, v, x, z    C, b, j, k, s, u, v, z    ph, j,
s, x, z    N, b, k, m, u, v, x, z    S, b, j, k, s, u, v, x, z    j, X    G, b, j, k, m, s, u, v, x, z    T, b, j, k, m, s, u, v, x, z    J, b, j, k, s, u, v, x, z    Q, b, k, m, u, v, x, z
Allowed substitution hints:    ph( v, u, k, m, b)    B( x, u, j, k, m, s)    C( x, m)    P( j, s)    Q( j, s)    S( m)    J( m)    K( x, z, v, u, j, k, m, s, b)    L( x, v, u, j, k, m, s, b)    N( j, s)    X( x, z, v, u, k, m, s, b)

Proof of Theorem cvmliftlem13
StepHypRef Expression
1 cvmliftlem.1 . . . . . . 7  |-  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 ) ) ) ) } )
2 cvmliftlem.b . . . . . . 7  |-  B  = 
U. C
3 cvmliftlem.x . . . . . . 7  |-  X  = 
U. J
4 cvmliftlem.f . . . . . . 7  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
5 cvmliftlem.g . . . . . . 7  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
6 cvmliftlem.p . . . . . . 7  |-  ( ph  ->  P  e.  B )
7 cvmliftlem.e . . . . . . 7  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
8 cvmliftlem.n . . . . . . 7  |-  ( ph  ->  N  e.  NN )
9 cvmliftlem.t . . . . . . 7  |-  ( ph  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
10 cvmliftlem.a . . . . . . 7  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
11 cvmliftlem.l . . . . . . 7  |-  L  =  ( topGen `  ran  (,) )
12 cvmliftlem.q . . . . . . 7  |-  Q  =  seq  0 ( ( x  e.  _V ,  m  e.  NN  |->  ( z  e.  ( ( ( m  -  1 )  /  N ) [,] ( m  /  N
) )  |->  ( `' ( F  |`  ( iota_ b  e.  ( 2nd `  ( T `  m
) ) ( x `
 ( ( m  -  1 )  /  N ) )  e.  b ) ) `  ( G `  z ) ) ) ) ,  ( (  _I  |`  NN )  u.  { <. 0 ,  { <. 0 ,  P >. } >. } ) )
13 cvmliftlem.k . . . . . . 7  |-  K  = 
U_ k  e.  ( 1 ... N ) ( Q `  k
)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13cvmliftlem11 24982 . . . . . 6  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
1514simpld 446 . . . . 5  |-  ( ph  ->  K  e.  ( II 
Cn  C ) )
16 iiuni 18911 . . . . . 6  |-  ( 0 [,] 1 )  = 
U. II
1716, 2cnf 17310 . . . . 5  |-  ( K  e.  ( II  Cn  C )  ->  K : ( 0 [,] 1 ) --> B )
1815, 17syl 16 . . . 4  |-  ( ph  ->  K : ( 0 [,] 1 ) --> B )
19 ffun 5593 . . . 4  |-  ( K : ( 0 [,] 1 ) --> B  ->  Fun  K )
2018, 19syl 16 . . 3  |-  ( ph  ->  Fun  K )
21 nnuz 10521 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
228, 21syl6eleq 2526 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
23 eluzfz1 11064 . . . . . 6  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
2422, 23syl 16 . . . . 5  |-  ( ph  ->  1  e.  ( 1 ... N ) )
25 fveq2 5728 . . . . . 6  |-  ( k  =  1  ->  ( Q `  k )  =  ( Q ` 
1 ) )
2625ssiun2s 4135 . . . . 5  |-  ( 1  e.  ( 1 ... N )  ->  ( Q `  1 )  C_ 
U_ k  e.  ( 1 ... N ) ( Q `  k
) )
2724, 26syl 16 . . . 4  |-  ( ph  ->  ( Q `  1
)  C_  U_ k  e.  ( 1 ... N
) ( Q `  k ) )
2827, 13syl6sseqr 3395 . . 3  |-  ( ph  ->  ( Q `  1
)  C_  K )
29 0xr 9131 . . . . . . 7  |-  0  e.  RR*
3029a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR* )
318nnrecred 10045 . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  e.  RR )
3231rexrd 9134 . . . . . 6  |-  ( ph  ->  ( 1  /  N
)  e.  RR* )
33 1re 9090 . . . . . . . 8  |-  1  e.  RR
3433a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
35 0le1 9551 . . . . . . . 8  |-  0  <_  1
3635a1i 11 . . . . . . 7  |-  ( ph  ->  0  <_  1 )
378nnred 10015 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
388nngt0d 10043 . . . . . . 7  |-  ( ph  ->  0  <  N )
39 divge0 9879 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( 1  /  N ) )
4034, 36, 37, 38, 39syl22anc 1185 . . . . . 6  |-  ( ph  ->  0  <_  ( 1  /  N ) )
41 lbicc2 11013 . . . . . 6  |-  ( ( 0  e.  RR*  /\  (
1  /  N )  e.  RR*  /\  0  <_  ( 1  /  N
) )  ->  0  e.  ( 0 [,] (
1  /  N ) ) )
4230, 32, 40, 41syl3anc 1184 . . . . 5  |-  ( ph  ->  0  e.  ( 0 [,] ( 1  /  N ) ) )
43 1m1e0 10068 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4443oveq1i 6091 . . . . . . 7  |-  ( ( 1  -  1 )  /  N )  =  ( 0  /  N
)
458nncnd 10016 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
468nnne0d 10044 . . . . . . . 8  |-  ( ph  ->  N  =/=  0 )
4745, 46div0d 9789 . . . . . . 7  |-  ( ph  ->  ( 0  /  N
)  =  0 )
4844, 47syl5eq 2480 . . . . . 6  |-  ( ph  ->  ( ( 1  -  1 )  /  N
)  =  0 )
4948oveq1d 6096 . . . . 5  |-  ( ph  ->  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) )  =  ( 0 [,] ( 1  /  N ) ) )
5042, 49eleqtrrd 2513 . . . 4  |-  ( ph  ->  0  e.  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) )
51 eqid 2436 . . . . . . . 8  |-  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) )  =  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) )
52 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  1  e.  ( 1 ... N
) )
531, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 51cvmliftlem7 24978 . . . . . . . 8  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  (
( Q `  (
1  -  1 ) ) `  ( ( 1  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( 1  -  1 )  /  N
) ) } ) )
541, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 51, 52, 53cvmliftlem6 24977 . . . . . . 7  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  (
( Q `  1
) : ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) --> B  /\  ( F  o.  ( Q `  1 ) )  =  ( G  |`  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) ) ) ) )
5524, 54mpdan 650 . . . . . 6  |-  ( ph  ->  ( ( Q ` 
1 ) : ( ( ( 1  -  1 )  /  N
) [,] ( 1  /  N ) ) --> B  /\  ( F  o.  ( Q ` 
1 ) )  =  ( G  |`  (
( ( 1  -  1 )  /  N
) [,] ( 1  /  N ) ) ) ) )
5655simpld 446 . . . . 5  |-  ( ph  ->  ( Q `  1
) : ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) --> B )
57 fdm 5595 . . . . 5  |-  ( ( Q `  1 ) : ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N
) ) --> B  ->  dom  ( Q `  1
)  =  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) )
5856, 57syl 16 . . . 4  |-  ( ph  ->  dom  ( Q ` 
1 )  =  ( ( ( 1  -  1 )  /  N
) [,] ( 1  /  N ) ) )
5950, 58eleqtrrd 2513 . . 3  |-  ( ph  ->  0  e.  dom  ( Q `  1 )
)
60 funssfv 5746 . . 3  |-  ( ( Fun  K  /\  ( Q `  1 )  C_  K  /\  0  e. 
dom  ( Q ` 
1 ) )  -> 
( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
6120, 28, 59, 60syl3anc 1184 . 2  |-  ( ph  ->  ( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
621, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem9 24980 . . . 4  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  (
( Q `  1
) `  ( (
1  -  1 )  /  N ) )  =  ( ( Q `
 ( 1  -  1 ) ) `  ( ( 1  -  1 )  /  N
) ) )
6324, 62mpdan 650 . . 3  |-  ( ph  ->  ( ( Q ` 
1 ) `  (
( 1  -  1 )  /  N ) )  =  ( ( Q `  ( 1  -  1 ) ) `
 ( ( 1  -  1 )  /  N ) ) )
6448fveq2d 5732 . . 3  |-  ( ph  ->  ( ( Q ` 
1 ) `  (
( 1  -  1 )  /  N ) )  =  ( ( Q `  1 ) `
 0 ) )
6543fveq2i 5731 . . . . . 6  |-  ( Q `
 ( 1  -  1 ) )  =  ( Q `  0
)
661, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem4 24975 . . . . . 6  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
6765, 66eqtri 2456 . . . . 5  |-  ( Q `
 ( 1  -  1 ) )  =  { <. 0 ,  P >. }
6867a1i 11 . . . 4  |-  ( ph  ->  ( Q `  (
1  -  1 ) )  =  { <. 0 ,  P >. } )
6968, 48fveq12d 5734 . . 3  |-  ( ph  ->  ( ( Q `  ( 1  -  1 ) ) `  (
( 1  -  1 )  /  N ) )  =  ( {
<. 0 ,  P >. } `  0 ) )
7063, 64, 693eqtr3d 2476 . 2  |-  ( ph  ->  ( ( Q ` 
1 ) `  0
)  =  ( {
<. 0 ,  P >. } `  0 ) )
71 0nn0 10236 . . 3  |-  0  e.  NN0
72 fvsng 5927 . . 3  |-  ( ( 0  e.  NN0  /\  P  e.  B )  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7371, 6, 72sylancr 645 . 2  |-  ( ph  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7461, 70, 733eqtrd 2472 1  |-  ( ph  ->  ( K `  0
)  =  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    \ cdif 3317    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   <.cop 3817   U.cuni 4015   U_ciun 4093   class class class wbr 4212    e. cmpt 4266    _I cid 4493    X. cxp 4876   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881    o. ccom 4882   Fun wfun 5448   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   iota_crio 6542   RRcr 8989   0cc0 8990   1c1 8991   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291    / cdiv 9677   NNcn 10000   NN0cn0 10221   ZZ>=cuz 10488   (,)cioo 10916   [,]cicc 10919   ...cfz 11043    seq cseq 11323   ↾t crest 13648   topGenctg 13665    Cn ccn 17288    Homeo chmeo 17785   IIcii 18905   CovMap ccvm 24942
This theorem is referenced by:  cvmliftlem14  24984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-icc 10923  df-fz 11044  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-rest 13650  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-cld 17083  df-cn 17291  df-hmeo 17787  df-ii 18907  df-cvm 24943
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