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Theorem cvmlift2lem6 27111
Description: Lemma for cvmlift2 27119. (Contributed by Mario Carneiro, 7-May-2015.)
Hypotheses
Ref Expression
cvmlift2.b  |-  B  = 
U. C
cvmlift2.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmlift2.g  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
cvmlift2.p  |-  ( ph  ->  P  e.  B )
cvmlift2.i  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
cvmlift2.h  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
cvmlift2.k  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
Assertion
Ref Expression
cvmlift2lem6  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  ( K  |`  ( { X }  X.  ( 0 [,] 1 ) ) )  e.  ( ( ( II  tX  II )t  ( { X }  X.  (
0 [,] 1 ) ) )  Cn  C
) )
Distinct variable groups:    x, f,
y, z, F    ph, f, x, y, z    f, J, x, y, z    f, G, x, y, z    f, H, x, y, z    f, X, x, y, z    C, f, x, y, z    P, f, x, y, z    x, B, y, z    f, K, x, y, z
Allowed substitution hint:    B( f)

Proof of Theorem cvmlift2lem6
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmlift2.b . . . . . . . 8  |-  B  = 
U. C
2 cvmlift2.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
3 cvmlift2.g . . . . . . . 8  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
4 cvmlift2.p . . . . . . . 8  |-  ( ph  ->  P  e.  B )
5 cvmlift2.i . . . . . . . 8  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
6 cvmlift2.h . . . . . . . 8  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
7 cvmlift2.k . . . . . . . 8  |-  K  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) )  /\  (
f `  0 )  =  ( H `  x ) ) ) `
 y ) )
81, 2, 3, 4, 5, 6, 7cvmlift2lem5 27110 . . . . . . 7  |-  ( ph  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
98adantr 462 . . . . . 6  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
10 ffn 5556 . . . . . 6  |-  ( K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B  ->  K  Fn  ( (
0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
119, 10syl 16 . . . . 5  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  K  Fn  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
12 fnov 6197 . . . . 5  |-  ( K  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  K  =  ( u  e.  (
0 [,] 1 ) ,  v  e.  ( 0 [,] 1 ) 
|->  ( u K v ) ) )
1311, 12sylib 196 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  K  =  ( u  e.  ( 0 [,] 1
) ,  v  e.  ( 0 [,] 1
)  |->  ( u K v ) ) )
1413reseq1d 5105 . . 3  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  ( K  |`  ( { X }  X.  ( 0 [,] 1 ) ) )  =  ( ( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  ( u K v ) )  |`  ( { X }  X.  ( 0 [,] 1
) ) ) )
15 simpr 458 . . . . . 6  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  X  e.  ( 0 [,] 1
) )
1615snssd 4015 . . . . 5  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  { X }  C_  ( 0 [,] 1 ) )
17 ssid 3372 . . . . 5  |-  ( 0 [,] 1 )  C_  ( 0 [,] 1
)
18 resmpt2 6187 . . . . 5  |-  ( ( { X }  C_  ( 0 [,] 1
)  /\  ( 0 [,] 1 )  C_  ( 0 [,] 1
) )  ->  (
( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 ) 
|->  ( u K v ) )  |`  ( { X }  X.  (
0 [,] 1 ) ) )  =  ( u  e.  { X } ,  v  e.  ( 0 [,] 1
)  |->  ( u K v ) ) )
1916, 17, 18sylancl 657 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 ) 
|->  ( u K v ) )  |`  ( { X }  X.  (
0 [,] 1 ) ) )  =  ( u  e.  { X } ,  v  e.  ( 0 [,] 1
)  |->  ( u K v ) ) )
20 elsni 3899 . . . . . . . 8  |-  ( u  e.  { X }  ->  u  =  X )
21203ad2ant2 1005 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  u  =  X )
2221oveq1d 6105 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u K v )  =  ( X K v ) )
23 simp1r 1008 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  X  e.  ( 0 [,] 1 ) )
24 simp3 985 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  v  e.  ( 0 [,] 1 ) )
251, 2, 3, 4, 5, 6, 7cvmlift2lem4 27109 . . . . . . 7  |-  ( ( X  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  ->  ( X K v )  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  v ) )
2623, 24, 25syl2anc 656 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  ( X K v )  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  v ) )
2722, 26eqtrd 2473 . . . . 5  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u K v )  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  v ) )
2827mpt2eq3dva 6149 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
u  e.  { X } ,  v  e.  ( 0 [,] 1
)  |->  ( u K v ) )  =  ( u  e.  { X } ,  v  e.  ( 0 [,] 1
)  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( X G z ) )  /\  ( f `  0
)  =  ( H `
 X ) ) ) `  v ) ) )
2919, 28eqtrd 2473 . . 3  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
( u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 ) 
|->  ( u K v ) )  |`  ( { X }  X.  (
0 [,] 1 ) ) )  =  ( u  e.  { X } ,  v  e.  ( 0 [,] 1
)  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( X G z ) )  /\  ( f `  0
)  =  ( H `
 X ) ) ) `  v ) ) )
3014, 29eqtrd 2473 . 2  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  ( K  |`  ( { X }  X.  ( 0 [,] 1 ) ) )  =  ( u  e. 
{ X } , 
v  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  v ) ) )
31 eqid 2441 . . . 4  |-  ( IIt  { X } )  =  ( IIt 
{ X } )
32 iitopon 20355 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3332a1i 11 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
34 eqid 2441 . . . 4  |-  ( IIt  ( 0 [,] 1 ) )  =  ( IIt  ( 0 [,] 1 ) )
3517a1i 11 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
0 [,] 1 ) 
C_  ( 0 [,] 1 ) )
3633, 33cnmpt2nd 19142 . . . . 5  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  v )  e.  ( ( II  tX  II )  Cn  II ) )
37 eqid 2441 . . . . . . 7  |-  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( X G z ) )  /\  ( f `  0
)  =  ( H `
 X ) ) )  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( X G z ) )  /\  ( f `  0
)  =  ( H `
 X ) ) )
381, 2, 3, 4, 5, 6, 37cvmlift2lem3 27108 . . . . . 6  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) )  e.  ( II  Cn  C
)  /\  ( F  o.  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( X G z ) )  /\  ( ( iota_ f  e.  ( II  Cn  C
) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 ) 
|->  ( X G z ) )  /\  (
f `  0 )  =  ( H `  X ) ) ) `
 0 )  =  ( H `  X
) ) )
3938simp1d 995 . . . . 5  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  ( iota_ f  e.  ( II 
Cn  C ) ( ( F  o.  f
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f ` 
0 )  =  ( H `  X ) ) )  e.  ( II  Cn  C ) )
4033, 33, 36, 39cnmpt21f 19145 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( X G z ) )  /\  ( f `
 0 )  =  ( H `  X
) ) ) `  v ) )  e.  ( ( II  tX  II )  Cn  C
) )
4131, 33, 16, 34, 33, 35, 40cnmpt2res 19150 . . 3  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
u  e.  { X } ,  v  e.  ( 0 [,] 1
)  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( X G z ) )  /\  ( f `  0
)  =  ( H `
 X ) ) ) `  v ) )  e.  ( ( ( IIt  { X } ) 
tX  ( IIt  ( 0 [,] 1 ) ) )  Cn  C ) )
42 iitop 20356 . . . . 5  |-  II  e.  Top
43 snex 4530 . . . . 5  |-  { X }  e.  _V
44 ovex 6115 . . . . 5  |-  ( 0 [,] 1 )  e. 
_V
45 txrest 19104 . . . . 5  |-  ( ( ( II  e.  Top  /\  II  e.  Top )  /\  ( { X }  e.  _V  /\  ( 0 [,] 1 )  e. 
_V ) )  -> 
( ( II  tX  II )t  ( { X }  X.  ( 0 [,] 1 ) ) )  =  ( ( IIt  { X } )  tX  (
IIt 
( 0 [,] 1
) ) ) )
4642, 42, 43, 44, 45mp4an 668 . . . 4  |-  ( ( II  tX  II )t  ( { X }  X.  (
0 [,] 1 ) ) )  =  ( ( IIt  { X } ) 
tX  ( IIt  ( 0 [,] 1 ) ) )
4746oveq1i 6100 . . 3  |-  ( ( ( II  tX  II )t  ( { X }  X.  ( 0 [,] 1
) ) )  Cn  C )  =  ( ( ( IIt  { X } )  tX  (
IIt 
( 0 [,] 1
) ) )  Cn  C )
4841, 47syl6eleqr 2532 . 2  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
u  e.  { X } ,  v  e.  ( 0 [,] 1
)  |->  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( X G z ) )  /\  ( f `  0
)  =  ( H `
 X ) ) ) `  v ) )  e.  ( ( ( II  tX  II )t  ( { X }  X.  ( 0 [,] 1
) ) )  Cn  C ) )
4930, 48eqeltrd 2515 1  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  ( K  |`  ( { X }  X.  ( 0 [,] 1 ) ) )  e.  ( ( ( II  tX  II )t  ( { X }  X.  (
0 [,] 1 ) ) )  Cn  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   {csn 3874   U.cuni 4088    e. cmpt 4347    X. cxp 4834    |` cres 4838    o. ccom 4840    Fn wfn 5410   -->wf 5411   ` cfv 5415   iota_crio 6048  (class class class)co 6090    e. cmpt2 6092   0cc0 9278   1c1 9279   [,]cicc 11299   ↾t crest 14355   Topctop 18398  TopOnctopon 18399    Cn ccn 18728    tX ctx 19033   IIcii 20351   CovMap ccvm 27058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-cn 18731  df-cnp 18732  df-cmp 18890  df-con 18916  df-lly 18970  df-nlly 18971  df-tx 19035  df-hmeo 19228  df-xms 19795  df-ms 19796  df-tms 19797  df-ii 20353  df-htpy 20442  df-phtpy 20443  df-phtpc 20464  df-pcon 27024  df-scon 27025  df-cvm 27059
This theorem is referenced by:  cvmlift2lem9  27114
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