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Theorem cvmliftlem13 29807
Description: Lemma for cvmlift 29810. 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 29806 . . . . . 6  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
1514simpld 460 . . . . 5  |-  ( ph  ->  K  e.  ( II 
Cn  C ) )
16 iiuni 21809 . . . . . 6  |-  ( 0 [,] 1 )  = 
U. II
1716, 2cnf 20193 . . . . 5  |-  ( K  e.  ( II  Cn  C )  ->  K : ( 0 [,] 1 ) --> B )
1815, 17syl 17 . . . 4  |-  ( ph  ->  K : ( 0 [,] 1 ) --> B )
19 ffun 5748 . . . 4  |-  ( K : ( 0 [,] 1 ) --> B  ->  Fun  K )
2018, 19syl 17 . . 3  |-  ( ph  ->  Fun  K )
21 nnuz 11194 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
228, 21syl6eleq 2527 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
23 eluzfz1 11804 . . . . . 6  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
2422, 23syl 17 . . . . 5  |-  ( ph  ->  1  e.  ( 1 ... N ) )
25 fveq2 5881 . . . . . 6  |-  ( k  =  1  ->  ( Q `  k )  =  ( Q ` 
1 ) )
2625ssiun2s 4346 . . . . 5  |-  ( 1  e.  ( 1 ... N )  ->  ( Q `  1 )  C_ 
U_ k  e.  ( 1 ... N ) ( Q `  k
) )
2724, 26syl 17 . . . 4  |-  ( ph  ->  ( Q `  1
)  C_  U_ k  e.  ( 1 ... N
) ( Q `  k ) )
2827, 13syl6sseqr 3517 . . 3  |-  ( ph  ->  ( Q `  1
)  C_  K )
29 0xr 9686 . . . . . . 7  |-  0  e.  RR*
3029a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR* )
318nnrecred 10655 . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  e.  RR )
3231rexrd 9689 . . . . . 6  |-  ( ph  ->  ( 1  /  N
)  e.  RR* )
33 1red 9657 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
34 0le1 10136 . . . . . . . 8  |-  0  <_  1
3534a1i 11 . . . . . . 7  |-  ( ph  ->  0  <_  1 )
368nnred 10624 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
378nngt0d 10653 . . . . . . 7  |-  ( ph  ->  0  <  N )
38 divge0 10473 . . . . . . 7  |-  ( ( ( 1  e.  RR  /\  0  <_  1 )  /\  ( N  e.  RR  /\  0  < 
N ) )  -> 
0  <_  ( 1  /  N ) )
3933, 35, 36, 37, 38syl22anc 1265 . . . . . 6  |-  ( ph  ->  0  <_  ( 1  /  N ) )
40 lbicc2 11746 . . . . . 6  |-  ( ( 0  e.  RR*  /\  (
1  /  N )  e.  RR*  /\  0  <_  ( 1  /  N
) )  ->  0  e.  ( 0 [,] (
1  /  N ) ) )
4130, 32, 39, 40syl3anc 1264 . . . . 5  |-  ( ph  ->  0  e.  ( 0 [,] ( 1  /  N ) ) )
42 1m1e0 10678 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4342oveq1i 6315 . . . . . . 7  |-  ( ( 1  -  1 )  /  N )  =  ( 0  /  N
)
448nncnd 10625 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
458nnne0d 10654 . . . . . . . 8  |-  ( ph  ->  N  =/=  0 )
4644, 45div0d 10381 . . . . . . 7  |-  ( ph  ->  ( 0  /  N
)  =  0 )
4743, 46syl5eq 2482 . . . . . 6  |-  ( ph  ->  ( ( 1  -  1 )  /  N
)  =  0 )
4847oveq1d 6320 . . . . 5  |-  ( ph  ->  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) )  =  ( 0 [,] ( 1  /  N ) ) )
4941, 48eleqtrrd 2520 . . . 4  |-  ( ph  ->  0  e.  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) )
50 eqid 2429 . . . . . . . 8  |-  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) )  =  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) )
51 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  1  e.  ( 1 ... N
) )
521, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 50cvmliftlem7 29802 . . . . . . . 8  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  (
( Q `  (
1  -  1 ) ) `  ( ( 1  -  1 )  /  N ) )  e.  ( `' F " { ( G `  ( ( 1  -  1 )  /  N
) ) } ) )
531, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 50, 51, 52cvmliftlem6 29801 . . . . . . 7  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  (
( Q `  1
) : ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) --> B  /\  ( F  o.  ( Q `  1 ) )  =  ( G  |`  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) ) ) ) )
5424, 53mpdan 672 . . . . . 6  |-  ( ph  ->  ( ( Q ` 
1 ) : ( ( ( 1  -  1 )  /  N
) [,] ( 1  /  N ) ) --> B  /\  ( F  o.  ( Q ` 
1 ) )  =  ( G  |`  (
( ( 1  -  1 )  /  N
) [,] ( 1  /  N ) ) ) ) )
5554simpld 460 . . . . 5  |-  ( ph  ->  ( Q `  1
) : ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) --> B )
56 fdm 5750 . . . . 5  |-  ( ( Q `  1 ) : ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N
) ) --> B  ->  dom  ( Q `  1
)  =  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) )
5755, 56syl 17 . . . 4  |-  ( ph  ->  dom  ( Q ` 
1 )  =  ( ( ( 1  -  1 )  /  N
) [,] ( 1  /  N ) ) )
5849, 57eleqtrrd 2520 . . 3  |-  ( ph  ->  0  e.  dom  ( Q `  1 )
)
59 funssfv 5896 . . 3  |-  ( ( Fun  K  /\  ( Q `  1 )  C_  K  /\  0  e. 
dom  ( Q ` 
1 ) )  -> 
( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
6020, 28, 58, 59syl3anc 1264 . 2  |-  ( ph  ->  ( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
611, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem9 29804 . . . 4  |-  ( (
ph  /\  1  e.  ( 1 ... N
) )  ->  (
( Q `  1
) `  ( (
1  -  1 )  /  N ) )  =  ( ( Q `
 ( 1  -  1 ) ) `  ( ( 1  -  1 )  /  N
) ) )
6224, 61mpdan 672 . . 3  |-  ( ph  ->  ( ( Q ` 
1 ) `  (
( 1  -  1 )  /  N ) )  =  ( ( Q `  ( 1  -  1 ) ) `
 ( ( 1  -  1 )  /  N ) ) )
6347fveq2d 5885 . . 3  |-  ( ph  ->  ( ( Q ` 
1 ) `  (
( 1  -  1 )  /  N ) )  =  ( ( Q `  1 ) `
 0 ) )
6442fveq2i 5884 . . . . . 6  |-  ( Q `
 ( 1  -  1 ) )  =  ( Q `  0
)
651, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem4 29799 . . . . . 6  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
6664, 65eqtri 2458 . . . . 5  |-  ( Q `
 ( 1  -  1 ) )  =  { <. 0 ,  P >. }
6766a1i 11 . . . 4  |-  ( ph  ->  ( Q `  (
1  -  1 ) )  =  { <. 0 ,  P >. } )
6867, 47fveq12d 5887 . . 3  |-  ( ph  ->  ( ( Q `  ( 1  -  1 ) ) `  (
( 1  -  1 )  /  N ) )  =  ( {
<. 0 ,  P >. } `  0 ) )
6962, 63, 683eqtr3d 2478 . 2  |-  ( ph  ->  ( ( Q ` 
1 ) `  0
)  =  ( {
<. 0 ,  P >. } `  0 ) )
70 0nn0 10884 . . 3  |-  0  e.  NN0
71 fvsng 6113 . . 3  |-  ( ( 0  e.  NN0  /\  P  e.  B )  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7270, 6, 71sylancr 667 . 2  |-  ( ph  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7360, 69, 723eqtrd 2474 1  |-  ( ph  ->  ( K `  0
)  =  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   _Vcvv 3087    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   {csn 4002   <.cop 4008   U.cuni 4222   U_ciun 4302   class class class wbr 4426    |-> cmpt 4484    _I cid 4764    X. cxp 4852   `'ccnv 4853   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857    o. ccom 4858   Fun wfun 5595   -->wf 5597   ` cfv 5601   iota_crio 6266  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   RRcr 9537   0cc0 9538   1c1 9539   RR*cxr 9673    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   NNcn 10609   NN0cn0 10869   ZZ>=cuz 11159   (,)cioo 11635   [,]cicc 11638   ...cfz 11782    seqcseq 12210   ↾t crest 15278   topGenctg 15295    Cn ccn 20171   Homeochmeo 20699   IIcii 21803   CovMap ccvm 29766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-rest 15280  df-topgen 15301  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-top 19852  df-bases 19853  df-topon 19854  df-cld 19965  df-cn 20174  df-hmeo 20701  df-ii 21805  df-cvm 29767
This theorem is referenced by:  cvmliftlem14  29808
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