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Theorem cvmliftlem11 24974
Description: Lemma for cvmlift 24978. (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 228 . . . . 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 24973 . . . 4  |-  ( ph  ->  ( K  e.  ( ( Lt  ( 0 [,] ( N  /  N
) ) )  Cn  C )  /\  ( F  o.  K )  =  ( G  |`  ( 0 [,] ( N  /  N ) ) ) ) )
1615simpld 446 . . 3  |-  ( ph  ->  K  e.  ( ( Lt  ( 0 [,] ( N  /  N ) ) )  Cn  C ) )
1711a1i 11 . . . . . 6  |-  ( ph  ->  L  =  ( topGen ` 
ran  (,) ) )
188nncnd 10008 . . . . . . . 8  |-  ( ph  ->  N  e.  CC )
198nnne0d 10036 . . . . . . . 8  |-  ( ph  ->  N  =/=  0 )
2018, 19dividd 9780 . . . . . . 7  |-  ( ph  ->  ( N  /  N
)  =  1 )
2120oveq2d 6089 . . . . . 6  |-  ( ph  ->  ( 0 [,] ( N  /  N ) )  =  ( 0 [,] 1 ) )
2217, 21oveq12d 6091 . . . . 5  |-  ( ph  ->  ( Lt  ( 0 [,] ( N  /  N
) ) )  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) ) )
23 dfii2 18904 . . . . 5  |-  II  =  ( ( topGen `  ran  (,) )t  ( 0 [,] 1
) )
2422, 23syl6eqr 2485 . . . 4  |-  ( ph  ->  ( Lt  ( 0 [,] ( N  /  N
) ) )  =  II )
2524oveq1d 6088 . . 3  |-  ( ph  ->  ( ( Lt  ( 0 [,] ( N  /  N ) ) )  Cn  C )  =  ( II  Cn  C
) )
2616, 25eleqtrd 2511 . 2  |-  ( ph  ->  K  e.  ( II 
Cn  C ) )
2715simprd 450 . . 3  |-  ( ph  ->  ( F  o.  K
)  =  ( G  |`  ( 0 [,] ( N  /  N ) ) ) )
2821reseq2d 5138 . . 3  |-  ( ph  ->  ( G  |`  (
0 [,] ( N  /  N ) ) )  =  ( G  |`  ( 0 [,] 1
) ) )
29 iiuni 18903 . . . . . 6  |-  ( 0 [,] 1 )  = 
U. II
3029, 3cnf 17302 . . . . 5  |-  ( G  e.  ( II  Cn  J )  ->  G : ( 0 [,] 1 ) --> X )
315, 30syl 16 . . . 4  |-  ( ph  ->  G : ( 0 [,] 1 ) --> X )
32 ffn 5583 . . . 4  |-  ( G : ( 0 [,] 1 ) --> X  ->  G  Fn  ( 0 [,] 1 ) )
33 fnresdm 5546 . . . 4  |-  ( G  Fn  ( 0 [,] 1 )  ->  ( G  |`  ( 0 [,] 1 ) )  =  G )
3431, 32, 333syl 19 . . 3  |-  ( ph  ->  ( G  |`  (
0 [,] 1 ) )  =  G )
3527, 28, 343eqtrd 2471 . 2  |-  ( ph  ->  ( F  o.  K
)  =  G )
3626, 35jca 519 1  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   {csn 3806   <.cop 3809   U.cuni 4007   U_ciun 4085    e. cmpt 4258    _I cid 4485    X. cxp 4868   `'ccnv 4869   ran crn 4871    |` cres 4872   "cima 4873    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   iota_crio 6534   0cc0 8982   1c1 8983    + caddc 8985    - cmin 9283    / cdiv 9669   NNcn 9992   (,)cioo 10908   [,]cicc 10911   ...cfz 11035    seq cseq 11315   ↾t crest 13640   topGenctg 13657    Cn ccn 17280    Homeo chmeo 17777   IIcii 18897   CovMap ccvm 24934
This theorem is referenced by:  cvmliftlem13  24975  cvmliftlem14  24976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-icc 10915  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-rest 13642  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-cld 17075  df-cn 17283  df-hmeo 17779  df-ii 18899  df-cvm 24935
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