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Theorem cvmliftlem13 24936
Description: Lemma for cvmlift 24939. 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 24935 . . . . . 6  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
1514simpld 446 . . . . 5  |-  ( ph  ->  K  e.  ( II 
Cn  C ) )
16 iiuni 18864 . . . . . 6  |-  ( 0 [,] 1 )  = 
U. II
1716, 2cnf 17264 . . . . 5  |-  ( K  e.  ( II  Cn  C )  ->  K : ( 0 [,] 1 ) --> B )
1815, 17syl 16 . . . 4  |-  ( ph  ->  K : ( 0 [,] 1 ) --> B )
19 ffun 5552 . . . 4  |-  ( K : ( 0 [,] 1 ) --> B  ->  Fun  K )
2018, 19syl 16 . . 3  |-  ( ph  ->  Fun  K )
21 nnuz 10477 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
228, 21syl6eleq 2494 . . . . . 6  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
23 eluzfz1 11020 . . . . . 6  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
2422, 23syl 16 . . . . 5  |-  ( ph  ->  1  e.  ( 1 ... N ) )
25 fveq2 5687 . . . . . 6  |-  ( k  =  1  ->  ( Q `  k )  =  ( Q ` 
1 ) )
2625ssiun2s 4095 . . . . 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 3355 . . 3  |-  ( ph  ->  ( Q `  1
)  C_  K )
29 0xr 9087 . . . . . . 7  |-  0  e.  RR*
3029a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR* )
318nnrecred 10001 . . . . . . 7  |-  ( ph  ->  ( 1  /  N
)  e.  RR )
3231rexrd 9090 . . . . . 6  |-  ( ph  ->  ( 1  /  N
)  e.  RR* )
33 1re 9046 . . . . . . . 8  |-  1  e.  RR
3433a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
35 0le1 9507 . . . . . . . 8  |-  0  <_  1
3635a1i 11 . . . . . . 7  |-  ( ph  ->  0  <_  1 )
378nnred 9971 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
388nngt0d 9999 . . . . . . 7  |-  ( ph  ->  0  <  N )
39 divge0 9835 . . . . . . 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 10969 . . . . . 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 10024 . . . . . . . 8  |-  ( 1  -  1 )  =  0
4443oveq1i 6050 . . . . . . 7  |-  ( ( 1  -  1 )  /  N )  =  ( 0  /  N
)
458nncnd 9972 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
468nnne0d 10000 . . . . . . . 8  |-  ( ph  ->  N  =/=  0 )
4745, 46div0d 9745 . . . . . . 7  |-  ( ph  ->  ( 0  /  N
)  =  0 )
4844, 47syl5eq 2448 . . . . . 6  |-  ( ph  ->  ( ( 1  -  1 )  /  N
)  =  0 )
4948oveq1d 6055 . . . . 5  |-  ( ph  ->  ( ( ( 1  -  1 )  /  N ) [,] (
1  /  N ) )  =  ( 0 [,] ( 1  /  N ) ) )
5042, 49eleqtrrd 2481 . . . 4  |-  ( ph  ->  0  e.  ( ( ( 1  -  1 )  /  N ) [,] ( 1  /  N ) ) )
51 eqid 2404 . . . . . . . 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 24931 . . . . . . . 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 24930 . . . . . . 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 5554 . . . . 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 2481 . . 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 1184 . 2  |-  ( ph  ->  ( K `  0
)  =  ( ( Q `  1 ) `
 0 ) )
621, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem9 24933 . . . 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 5691 . . 3  |-  ( ph  ->  ( ( Q ` 
1 ) `  (
( 1  -  1 )  /  N ) )  =  ( ( Q `  1 ) `
 0 ) )
6543fveq2i 5690 . . . . . 6  |-  ( Q `
 ( 1  -  1 ) )  =  ( Q `  0
)
661, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cvmliftlem4 24928 . . . . . 6  |-  ( Q `
 0 )  =  { <. 0 ,  P >. }
6765, 66eqtri 2424 . . . . 5  |-  ( Q `
 ( 1  -  1 ) )  =  { <. 0 ,  P >. }
6867a1i 11 . . . 4  |-  ( ph  ->  ( Q `  (
1  -  1 ) )  =  { <. 0 ,  P >. } )
6968, 48fveq12d 5693 . . 3  |-  ( ph  ->  ( ( Q `  ( 1  -  1 ) ) `  (
( 1  -  1 )  /  N ) )  =  ( {
<. 0 ,  P >. } `  0 ) )
7063, 64, 693eqtr3d 2444 . 2  |-  ( ph  ->  ( ( Q ` 
1 ) `  0
)  =  ( {
<. 0 ,  P >. } `  0 ) )
71 0nn0 10192 . . 3  |-  0  e.  NN0
72 fvsng 5886 . . 3  |-  ( ( 0  e.  NN0  /\  P  e.  B )  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7371, 6, 72sylancr 645 . 2  |-  ( ph  ->  ( { <. 0 ,  P >. } `  0
)  =  P )
7461, 70, 733eqtrd 2440 1  |-  ( ph  ->  ( K `  0
)  =  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   <.cop 3777   U.cuni 3975   U_ciun 4053   class class class wbr 4172    e. cmpt 4226    _I cid 4453    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840    o. ccom 4841   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   iota_crio 6501   RRcr 8945   0cc0 8946   1c1 8947   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZ>=cuz 10444   (,)cioo 10872   [,]cicc 10875   ...cfz 10999    seq cseq 11278   ↾t crest 13603   topGenctg 13620    Cn ccn 17242    Homeo chmeo 17738   IIcii 18858   CovMap ccvm 24895
This theorem is referenced by:  cvmliftlem14  24937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-rest 13605  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038  df-cn 17245  df-hmeo 17740  df-ii 18860  df-cvm 24896
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