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Theorem cvmliftlem13 27185
Description: Lemma for cvmlift 27188. 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 27184 . . . . . 6  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
1514simpld 459 . . . . 5  |-  ( ph  ->  K  e.  ( II 
Cn  C ) )
16 iiuni 20457 . . . . . 6  |-  ( 0 [,] 1 )  = 
U. II
1716, 2cnf 18850 . . . . 5  |-  ( K  e.  ( II  Cn  C )  ->  K : ( 0 [,] 1 ) --> B )
1815, 17syl 16 . . . 4  |-  ( ph  ->  K : ( 0 [,] 1 ) --> B )
19 ffun 5561 . . . 4  |-  ( K : ( 0 [,] 1 ) --> B  ->  Fun  K )
2018, 19syl 16 . . 3  |-  ( ph  ->  Fun  K )
21 nnuz 10896 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
228, 21syl6eleq 2533 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
23 eluzfz1 11458 . . . . . 6  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
2422, 23syl 16 . . . . 5  |-  ( ph  ->  1  e.  ( 1 ... N ) )
25 fveq2 5691 . . . . . 6  |-  ( k  =  1  ->  ( Q `  k )  =  ( Q ` 
1 ) )
2625ssiun2s 4214 . . . . 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 3403 . . 3  |-  ( ph  ->  ( Q `  1
)  C_  K )
29 0xr 9430 . . . . . . 7  |-  0  e.  RR*
3029a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR* )
318nnrecred 10367 . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  e.  RR )
3231rexrd 9433 . . . . . 6  |-  ( ph  ->  ( 1  /  N
)  e.  RR* )
33 1re 9385 . . . . . . . 8  |-  1  e.  RR
3433a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
35 0le1 9863 . . . . . . . 8  |-  0  <_  1
3635a1i 11 . . . . . . 7  |-  ( ph  ->  0  <_  1 )
378nnred 10337 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
388nngt0d 10365 . . . . . . 7  |-  ( ph  ->  0  <  N )
39 divge0 10198 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( 1  /  N ) )
4034, 36, 37, 38, 39syl22anc 1219 . . . . . 6  |-  ( ph  ->  0  <_  ( 1  /  N ) )
41 lbicc2 11401 . . . . . 6  |-  ( ( 0  e.  RR*  /\  (
1  /  N )  e.  RR*  /\  0  <_  ( 1  /  N
) )  ->  0  e.  ( 0 [,] (
1  /  N ) ) )
4230, 32, 40, 41syl3anc 1218 . . . . 5  |-  ( ph  ->  0  e.  ( 0 [,] ( 1  /  N ) ) )
43 1m1e0 10390 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4443oveq1i 6101 . . . . . . 7  |-  ( ( 1  -  1 )  /  N )  =  ( 0  /  N
)
458nncnd 10338 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
468nnne0d 10366 . . . . . . . 8  |-  ( ph  ->  N  =/=  0 )
4745, 46div0d 10106 . . . . . . 7  |-  ( ph  ->  ( 0  /  N
)  =  0 )
4844, 47syl5eq 2487 . . . . . 6  |-  ( ph  ->  ( ( 1  -  1 )  /  N
)  =  0 )
4948oveq1d 6106 . . . . 5  |-  ( ph  ->  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) )  =  ( 0 [,] ( 1  /  N ) ) )
5042, 49eleqtrrd 2520 . . . 4  |-  ( ph  ->  0  e.  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) )
51 eqid 2443 . . . . . . . 8  |-  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) )  =  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) )
52 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  1  e.  ( 1 ... N
) )
531, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 51cvmliftlem7 27180 . . . . . . . 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 27179 . . . . . . 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 668 . . . . . 6  |-  ( ph  ->  ( ( Q ` 
1 ) : ( ( ( 1  -  1 )  /  N
) [,] ( 1  /  N ) ) --> B  /\  ( F  o.  ( Q ` 
1 ) )  =  ( G  |`  (
( ( 1  -  1 )  /  N
) [,] ( 1  /  N ) ) ) ) )
5655simpld 459 . . . . 5  |-  ( ph  ->  ( Q `  1
) : ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) --> B )
57 fdm 5563 . . . . 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 2520 . . 3  |-  ( ph  ->  0  e.  dom  ( Q `  1 )
)
60 funssfv 5705 . . 3  |-  ( ( Fun  K  /\  ( Q `  1 )  C_  K  /\  0  e. 
dom  ( Q ` 
1 ) )  -> 
( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
6120, 28, 59, 60syl3anc 1218 . 2  |-  ( ph  ->  ( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
621, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem9 27182 . . . 4  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  (
( Q `  1
) `  ( (
1  -  1 )  /  N ) )  =  ( ( Q `
 ( 1  -  1 ) ) `  ( ( 1  -  1 )  /  N
) ) )
6324, 62mpdan 668 . . 3  |-  ( ph  ->  ( ( Q ` 
1 ) `  (
( 1  -  1 )  /  N ) )  =  ( ( Q `  ( 1  -  1 ) ) `
 ( ( 1  -  1 )  /  N ) ) )
6448fveq2d 5695 . . 3  |-  ( ph  ->  ( ( Q ` 
1 ) `  (
( 1  -  1 )  /  N ) )  =  ( ( Q `  1 ) `
 0 ) )
6543fveq2i 5694 . . . . . 6  |-  ( Q `
 ( 1  -  1 ) )  =  ( Q `  0
)
661, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem4 27177 . . . . . 6  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
6765, 66eqtri 2463 . . . . 5  |-  ( Q `
 ( 1  -  1 ) )  =  { <. 0 ,  P >. }
6867a1i 11 . . . 4  |-  ( ph  ->  ( Q `  (
1  -  1 ) )  =  { <. 0 ,  P >. } )
6968, 48fveq12d 5697 . . 3  |-  ( ph  ->  ( ( Q `  ( 1  -  1 ) ) `  (
( 1  -  1 )  /  N ) )  =  ( {
<. 0 ,  P >. } `  0 ) )
7063, 64, 693eqtr3d 2483 . 2  |-  ( ph  ->  ( ( Q ` 
1 ) `  0
)  =  ( {
<. 0 ,  P >. } `  0 ) )
71 0nn0 10594 . . 3  |-  0  e.  NN0
72 fvsng 5912 . . 3  |-  ( ( 0  e.  NN0  /\  P  e.  B )  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7371, 6, 72sylancr 663 . 2  |-  ( ph  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7461, 70, 733eqtrd 2479 1  |-  ( ph  ->  ( K `  0
)  =  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   _Vcvv 2972    \ cdif 3325    u. cun 3326    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   {csn 3877   <.cop 3883   U.cuni 4091   U_ciun 4171   class class class wbr 4292    e. cmpt 4350    _I cid 4631    X. cxp 4838   `'ccnv 4839   dom cdm 4840   ran crn 4841    |` cres 4842   "cima 4843    o. ccom 4844   Fun wfun 5412   -->wf 5414   ` cfv 5418   iota_crio 6051  (class class class)co 6091    e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576   RRcr 9281   0cc0 9282   1c1 9283   RR*cxr 9417    < clt 9418    <_ cle 9419    - cmin 9595    / cdiv 9993   NNcn 10322   NN0cn0 10579   ZZ>=cuz 10861   (,)cioo 11300   [,]cicc 11303   ...cfz 11437    seqcseq 11806   ↾t crest 14359   topGenctg 14376    Cn ccn 18828   Homeochmeo 19326   IIcii 20451   CovMap ccvm 27144
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-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fi 7661  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-icc 11307  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-rest 14361  df-topgen 14382  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-top 18503  df-bases 18505  df-topon 18506  df-cld 18623  df-cn 18831  df-hmeo 19328  df-ii 20453  df-cvm 27145
This theorem is referenced by:  cvmliftlem14  27186
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