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Theorem cvmliftlem13 28397
Description: Lemma for cvmlift 28400. 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 28396 . . . . . 6  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
1514simpld 459 . . . . 5  |-  ( ph  ->  K  e.  ( II 
Cn  C ) )
16 iiuni 21136 . . . . . 6  |-  ( 0 [,] 1 )  = 
U. II
1716, 2cnf 19529 . . . . 5  |-  ( K  e.  ( II  Cn  C )  ->  K : ( 0 [,] 1 ) --> B )
1815, 17syl 16 . . . 4  |-  ( ph  ->  K : ( 0 [,] 1 ) --> B )
19 ffun 5732 . . . 4  |-  ( K : ( 0 [,] 1 ) --> B  ->  Fun  K )
2018, 19syl 16 . . 3  |-  ( ph  ->  Fun  K )
21 nnuz 11116 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
228, 21syl6eleq 2565 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
23 eluzfz1 11692 . . . . . 6  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
2422, 23syl 16 . . . . 5  |-  ( ph  ->  1  e.  ( 1 ... N ) )
25 fveq2 5865 . . . . . 6  |-  ( k  =  1  ->  ( Q `  k )  =  ( Q ` 
1 ) )
2625ssiun2s 4369 . . . . 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 3551 . . 3  |-  ( ph  ->  ( Q `  1
)  C_  K )
29 0xr 9639 . . . . . . 7  |-  0  e.  RR*
3029a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR* )
318nnrecred 10580 . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  e.  RR )
3231rexrd 9642 . . . . . 6  |-  ( ph  ->  ( 1  /  N
)  e.  RR* )
33 1re 9594 . . . . . . . 8  |-  1  e.  RR
3433a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
35 0le1 10075 . . . . . . . 8  |-  0  <_  1
3635a1i 11 . . . . . . 7  |-  ( ph  ->  0  <_  1 )
378nnred 10550 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
388nngt0d 10578 . . . . . . 7  |-  ( ph  ->  0  <  N )
39 divge0 10410 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( 1  /  N ) )
4034, 36, 37, 38, 39syl22anc 1229 . . . . . 6  |-  ( ph  ->  0  <_  ( 1  /  N ) )
41 lbicc2 11635 . . . . . 6  |-  ( ( 0  e.  RR*  /\  (
1  /  N )  e.  RR*  /\  0  <_  ( 1  /  N
) )  ->  0  e.  ( 0 [,] (
1  /  N ) ) )
4230, 32, 40, 41syl3anc 1228 . . . . 5  |-  ( ph  ->  0  e.  ( 0 [,] ( 1  /  N ) ) )
43 1m1e0 10603 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4443oveq1i 6293 . . . . . . 7  |-  ( ( 1  -  1 )  /  N )  =  ( 0  /  N
)
458nncnd 10551 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
468nnne0d 10579 . . . . . . . 8  |-  ( ph  ->  N  =/=  0 )
4745, 46div0d 10318 . . . . . . 7  |-  ( ph  ->  ( 0  /  N
)  =  0 )
4844, 47syl5eq 2520 . . . . . 6  |-  ( ph  ->  ( ( 1  -  1 )  /  N
)  =  0 )
4948oveq1d 6298 . . . . 5  |-  ( ph  ->  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) )  =  ( 0 [,] ( 1  /  N ) ) )
5042, 49eleqtrrd 2558 . . . 4  |-  ( ph  ->  0  e.  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) )
51 eqid 2467 . . . . . . . 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 28392 . . . . . . . 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 28391 . . . . . . 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 5734 . . . . 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 2558 . . 3  |-  ( ph  ->  0  e.  dom  ( Q `  1 )
)
60 funssfv 5880 . . 3  |-  ( ( Fun  K  /\  ( Q `  1 )  C_  K  /\  0  e. 
dom  ( Q ` 
1 ) )  -> 
( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
6120, 28, 59, 60syl3anc 1228 . 2  |-  ( ph  ->  ( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
621, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem9 28394 . . . 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 5869 . . 3  |-  ( ph  ->  ( ( Q ` 
1 ) `  (
( 1  -  1 )  /  N ) )  =  ( ( Q `  1 ) `
 0 ) )
6543fveq2i 5868 . . . . . 6  |-  ( Q `
 ( 1  -  1 ) )  =  ( Q `  0
)
661, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem4 28389 . . . . . 6  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
6765, 66eqtri 2496 . . . . 5  |-  ( Q `
 ( 1  -  1 ) )  =  { <. 0 ,  P >. }
6867a1i 11 . . . 4  |-  ( ph  ->  ( Q `  (
1  -  1 ) )  =  { <. 0 ,  P >. } )
6968, 48fveq12d 5871 . . 3  |-  ( ph  ->  ( ( Q `  ( 1  -  1 ) ) `  (
( 1  -  1 )  /  N ) )  =  ( {
<. 0 ,  P >. } `  0 ) )
7063, 64, 693eqtr3d 2516 . 2  |-  ( ph  ->  ( ( Q ` 
1 ) `  0
)  =  ( {
<. 0 ,  P >. } `  0 ) )
71 0nn0 10809 . . 3  |-  0  e.  NN0
72 fvsng 6094 . . 3  |-  ( ( 0  e.  NN0  /\  P  e.  B )  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7371, 6, 72sylancr 663 . 2  |-  ( ph  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7461, 70, 733eqtrd 2512 1  |-  ( ph  ->  ( K `  0
)  =  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   <.cop 4033   U.cuni 4245   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002    o. ccom 5003   Fun wfun 5581   -->wf 5583   ` cfv 5587   iota_crio 6243  (class class class)co 6283    |-> cmpt2 6285   1stc1st 6782   2ndc2nd 6783   RRcr 9490   0cc0 9491   1c1 9492   RR*cxr 9626    < clt 9627    <_ cle 9628    - cmin 9804    / cdiv 10205   NNcn 10535   NN0cn0 10794   ZZ>=cuz 11081   (,)cioo 11528   [,]cicc 11531   ...cfz 11671    seqcseq 12074   ↾t crest 14675   topGenctg 14692    Cn ccn 19507   Homeochmeo 20005   IIcii 21130   CovMap ccvm 28356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7870  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-icc 11535  df-fz 11672  df-seq 12075  df-exp 12134  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-rest 14677  df-topgen 14698  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-top 19182  df-bases 19184  df-topon 19185  df-cld 19302  df-cn 19510  df-hmeo 20007  df-ii 21132  df-cvm 28357
This theorem is referenced by:  cvmliftlem14  28398
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