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Theorem cvmliftlem11 23841
Description: Lemma for cvmlift 23845. (Contributed by Mario Carneiro, 14-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
cvmliftlem11  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
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 cvmliftlem11
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 cvmliftlem.1 . . . . 5  |-  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 . . . . 5  |-  B  = 
U. C
3 cvmliftlem.x . . . . 5  |-  X  = 
U. J
4 cvmliftlem.f . . . . 5  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
5 cvmliftlem.g . . . . 5  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
6 cvmliftlem.p . . . . 5  |-  ( ph  ->  P  e.  B )
7 cvmliftlem.e . . . . 5  |-  ( ph  ->  ( F `  P
)  =  ( G `
 0 ) )
8 cvmliftlem.n . . . . 5  |-  ( ph  ->  N  e.  NN )
9 cvmliftlem.t . . . . 5  |-  ( ph  ->  T : ( 1 ... N ) --> U_ j  e.  J  ( { j }  X.  ( S `  j ) ) )
10 cvmliftlem.a . . . . 5  |-  ( ph  ->  A. k  e.  ( 1 ... N ) ( G " (
( ( k  - 
1 )  /  N
) [,] ( k  /  N ) ) )  C_  ( 1st `  ( T `  k
) ) )
11 cvmliftlem.l . . . . 5  |-  L  =  ( topGen `  ran  (,) )
12 cvmliftlem.q . . . . 5  |-  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 . . . . 5  |-  K  = 
U_ k  e.  ( 1 ... N ) ( Q `  k
)
14 biid 227 . . . . 5  |-  ( ( ( n  e.  NN  /\  ( n  +  1 )  e.  ( 1 ... N ) )  /\  ( U_ k  e.  ( 1 ... n
) ( Q `  k )  e.  ( ( Lt  ( 0 [,] ( n  /  N
) ) )  Cn  C )  /\  ( F  o.  U_ k  e.  ( 1 ... n
) ( Q `  k ) )  =  ( G  |`  (
0 [,] ( n  /  N ) ) ) ) )  <->  ( (
n  e.  NN  /\  ( n  +  1
)  e.  ( 1 ... N ) )  /\  ( U_ k  e.  ( 1 ... n
) ( Q `  k )  e.  ( ( Lt  ( 0 [,] ( n  /  N
) ) )  Cn  C )  /\  ( F  o.  U_ k  e.  ( 1 ... n
) ( Q `  k ) )  =  ( G  |`  (
0 [,] ( n  /  N ) ) ) ) ) )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cvmliftlem10 23840 . . . 4  |-  ( ph  ->  ( K  e.  ( ( Lt  ( 0 [,] ( N  /  N
) ) )  Cn  C )  /\  ( F  o.  K )  =  ( G  |`  ( 0 [,] ( N  /  N ) ) ) ) )
1615simpld 445 . . 3  |-  ( ph  ->  K  e.  ( ( Lt  ( 0 [,] ( N  /  N ) ) )  Cn  C ) )
1711a1i 10 . . . . . 6  |-  ( ph  ->  L  =  ( topGen ` 
ran  (,) ) )
188nncnd 9778 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
198nnne0d 9806 . . . . . . . 8  |-  ( ph  ->  N  =/=  0 )
2018, 19dividd 9550 . . . . . . 7  |-  ( ph  ->  ( N  /  N
)  =  1 )
2120oveq2d 5890 . . . . . 6  |-  ( ph  ->  ( 0 [,] ( N  /  N ) )  =  ( 0 [,] 1 ) )
2217, 21oveq12d 5892 . . . . 5  |-  ( ph  ->  ( Lt  ( 0 [,] ( N  /  N
) ) )  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) ) )
23 dfii2 18402 . . . . 5  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
2422, 23syl6eqr 2346 . . . 4  |-  ( ph  ->  ( Lt  ( 0 [,] ( N  /  N
) ) )  =  II )
2524oveq1d 5889 . . 3  |-  ( ph  ->  ( ( Lt  ( 0 [,] ( N  /  N ) ) )  Cn  C )  =  ( II  Cn  C
) )
2616, 25eleqtrd 2372 . 2  |-  ( ph  ->  K  e.  ( II 
Cn  C ) )
2715simprd 449 . . 3  |-  ( ph  ->  ( F  o.  K
)  =  ( G  |`  ( 0 [,] ( N  /  N ) ) ) )
2821reseq2d 4971 . . 3  |-  ( ph  ->  ( G  |`  (
0 [,] ( N  /  N ) ) )  =  ( G  |`  ( 0 [,] 1
) ) )
29 iiuni 18401 . . . . . 6  |-  ( 0 [,] 1 )  = 
U. II
3029, 3cnf 16992 . . . . 5  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
315, 30syl 15 . . . 4  |-  ( ph  ->  G : ( 0 [,] 1 ) --> X )
32 ffn 5405 . . . 4  |-  ( G : ( 0 [,] 1 ) --> X  ->  G  Fn  ( 0 [,] 1 ) )
33 fnresdm 5369 . . . 4  |-  ( G  Fn  ( 0 [,] 1 )  ->  ( G  |`  ( 0 [,] 1 ) )  =  G )
3431, 32, 333syl 18 . . 3  |-  ( ph  ->  ( G  |`  (
0 [,] 1 ) )  =  G )
3527, 28, 343eqtrd 2332 . 2  |-  ( ph  ->  ( F  o.  K
)  =  G )
3626, 35jca 518 1  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   <.cop 3656   U.cuni 3843   U_ciun 3921    e. cmpt 4093    _I cid 4320    X. cxp 4703   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   iota_crio 6313   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053    / cdiv 9439   NNcn 9762   (,)cioo 10672   [,]cicc 10675   ...cfz 10798    seq cseq 11062   ↾t crest 13341   topGenctg 13358    Cn ccn 16970    Homeo chmeo 17460   IIcii 18395   CovMap ccvm 23801
This theorem is referenced by:  cvmliftlem13  23842  cvmliftlem14  23843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-rest 13343  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-cn 16973  df-hmeo 17462  df-ii 18397  df-cvm 23802
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