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Theorem cvmliftlem14 24763
Description: Lemma for cvmlift 24765. Putting the results of cvmliftlem11 24761, cvmliftlem13 24762 and cvmliftmo 24750 together, we have that  K is a continuous function, satisfies  F  o.  K  =  G and  K ( 0 )  =  P, and is equal to any other function which also has these properties, so it follows that  K is the unique lift of  G. (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
cvmliftlem14  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
Distinct variable groups:    v, b,
z, B    f, b,
j, k, m, s, u, x, F, v, z    z, L    f, K    P, b, f, k, m, u, v, x, z    C, b, f, j, k, s, u, v, z    ph, f, j, s, x, z    N, b, k, m, u, v, x, z    S, b, f, j, k, s, u, v, x, z   
j, X    G, b,
f, j, k, m, s, u, v, x, z    T, b, j, k, m, s, u, v, x, z    J, b, f, 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, f, j, k, m, s)    C( x, m)    P( j, s)    Q( f, j, s)    S( m)    T( f)    J( m)    K( x, z, v, u, j, k, m, s, b)    L( x, v, u, f, j, k, m, s, b)    N( f, j, s)    X( x, z, v, u, f, k, m, s, b)

Proof of Theorem cvmliftlem14
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
)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13cvmliftlem11 24761 . . . 4  |-  ( ph  ->  ( K  e.  ( II  Cn  C )  /\  ( F  o.  K )  =  G ) )
1514simpld 446 . . 3  |-  ( ph  ->  K  e.  ( II 
Cn  C ) )
1614simprd 450 . . 3  |-  ( ph  ->  ( F  o.  K
)  =  G )
171, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13cvmliftlem13 24762 . . 3  |-  ( ph  ->  ( K `  0
)  =  P )
18 coeq2 4971 . . . . . 6  |-  ( f  =  K  ->  ( F  o.  f )  =  ( F  o.  K ) )
1918eqeq1d 2395 . . . . 5  |-  ( f  =  K  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  K )  =  G ) )
20 fveq1 5667 . . . . . 6  |-  ( f  =  K  ->  (
f `  0 )  =  ( K ` 
0 ) )
2120eqeq1d 2395 . . . . 5  |-  ( f  =  K  ->  (
( f `  0
)  =  P  <->  ( K `  0 )  =  P ) )
2219, 21anbi12d 692 . . . 4  |-  ( f  =  K  ->  (
( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P )  <->  ( ( F  o.  K )  =  G  /\  ( K `
 0 )  =  P ) ) )
2322rspcev 2995 . . 3  |-  ( ( K  e.  ( II 
Cn  C )  /\  ( ( F  o.  K )  =  G  /\  ( K ` 
0 )  =  P ) )  ->  E. f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  G  /\  ( f `
 0 )  =  P ) )
2415, 16, 17, 23syl12anc 1182 . 2  |-  ( ph  ->  E. f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
25 iiuni 18782 . . 3  |-  ( 0 [,] 1 )  = 
U. II
26 iicon 18788 . . . 4  |-  II  e.  Con
2726a1i 11 . . 3  |-  ( ph  ->  II  e.  Con )
28 iinllycon 24720 . . . 4  |-  II  e. 𝑛Locally  Con
2928a1i 11 . . 3  |-  ( ph  ->  II  e. 𝑛Locally  Con )
30 0elunit 10947 . . . 4  |-  0  e.  ( 0 [,] 1
)
3130a1i 11 . . 3  |-  ( ph  ->  0  e.  ( 0 [,] 1 ) )
322, 25, 4, 27, 29, 31, 5, 6, 7cvmliftmo 24750 . 2  |-  ( ph  ->  E* f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
33 reu5 2864 . 2  |-  ( E! f  e.  ( II 
Cn  C ) ( ( F  o.  f
)  =  G  /\  ( f `  0
)  =  P )  <-> 
( E. f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  G  /\  ( f `
 0 )  =  P )  /\  E* f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  (
f `  0 )  =  P ) ) )
3424, 32, 33sylanbrc 646 1  |-  ( ph  ->  E! f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f ` 
0 )  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650   E!wreu 2651   E*wrmo 2652   {crab 2653   _Vcvv 2899    \ cdif 3260    u. cun 3261    i^i cin 3262    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   {csn 3757   <.cop 3760   U.cuni 3957   U_ciun 4035    e. cmpt 4207    _I cid 4434    X. cxp 4816   `'ccnv 4817   ran crn 4819    |` cres 4820   "cima 4821    o. ccom 4822   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286   2ndc2nd 6287   iota_crio 6478   0cc0 8923   1c1 8924    - cmin 9223    / cdiv 9609   NNcn 9932   (,)cioo 10848   [,]cicc 10851   ...cfz 10975    seq cseq 11250   ↾t crest 13575   topGenctg 13592    Cn ccn 17210   Conccon 17395  𝑛Locally cnlly 17449    Homeo chmeo 17706   IIcii 18776   CovMap ccvm 24721
This theorem is referenced by:  cvmliftlem15  24764
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-nei 17085  df-cn 17213  df-cnp 17214  df-con 17396  df-lly 17450  df-nlly 17451  df-tx 17515  df-hmeo 17708  df-xms 18259  df-ms 18260  df-tms 18261  df-ii 18778  df-htpy 18866  df-phtpy 18867  df-phtpc 18888  df-pcon 24687  df-scon 24688  df-cvm 24722
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