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Theorem cvmlift2lem6 24948
Description: Lemma for cvmlift2 24956. (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 24947 . . . . . . 7  |-  ( ph  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
98adantr 452 . . . . . 6  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
10 ffn 5550 . . . . . 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 6137 . . . . 5  |-  ( K  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  K  =  ( u  e.  (
0 [,] 1 ) ,  v  e.  ( 0 [,] 1 ) 
|->  ( u K v ) ) )
1311, 12sylib 189 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  K  =  ( u  e.  ( 0 [,] 1
) ,  v  e.  ( 0 [,] 1
)  |->  ( u K v ) ) )
1413reseq1d 5104 . . 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 448 . . . . . 6  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  X  e.  ( 0 [,] 1
) )
1615snssd 3903 . . . . 5  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  { X }  C_  ( 0 [,] 1 ) )
17 ssid 3327 . . . . 5  |-  ( 0 [,] 1 )  C_  ( 0 [,] 1
)
18 resmpt2 6127 . . . . 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 644 . . . 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 3798 . . . . . . . 8  |-  ( u  e.  { X }  ->  u  =  X )
21203ad2ant2 979 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  u  =  X )
2221oveq1d 6055 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u K v )  =  ( X K v ) )
23 simp1r 982 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  X  e.  ( 0 [,] 1 ) )
24 simp3 959 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  v  e.  ( 0 [,] 1 ) )
251, 2, 3, 4, 5, 6, 7cvmlift2lem4 24946 . . . . . . 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 643 . . . . . 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 2436 . . . . 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 6097 . . . 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 2436 . . 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 2436 . 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 2404 . . . 4  |-  ( IIt  { X } )  =  ( IIt 
{ X } )
32 iitopon 18862 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3332a1i 11 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
34 eqid 2404 . . . 4  |-  ( IIt  ( 0 [,] 1 ) )  =  ( IIt  ( 0 [,] 1 ) )
3517a1i 11 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
0 [,] 1 ) 
C_  ( 0 [,] 1 ) )
3633, 33cnmpt2nd 17654 . . . . 5  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  v )  e.  ( ( II  tX  II )  Cn  II ) )
37 eqid 2404 . . . . . . 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 24945 . . . . . 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 969 . . . . 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 17657 . . . 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 17662 . . 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 18863 . . . . 5  |-  II  e.  Top
43 snex 4365 . . . . 5  |-  { X }  e.  _V
44 ovex 6065 . . . . 5  |-  ( 0 [,] 1 )  e. 
_V
45 txrest 17616 . . . . 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 655 . . . 4  |-  ( ( II  tX  II )t  ( { X }  X.  (
0 [,] 1 ) ) )  =  ( ( IIt  { X } ) 
tX  ( IIt  ( 0 [,] 1 ) ) )
4746oveq1i 6050 . . 3  |-  ( ( ( II  tX  II )t  ( { X }  X.  ( 0 [,] 1
) ) )  Cn  C )  =  ( ( ( IIt  { X } )  tX  (
IIt 
( 0 [,] 1
) ) )  Cn  C )
4841, 47syl6eleqr 2495 . 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 2478 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   {csn 3774   U.cuni 3975    e. cmpt 4226    X. cxp 4835    |` cres 4839    o. ccom 4841    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   iota_crio 6501   0cc0 8946   1c1 8947   [,]cicc 10875   ↾t crest 13603   Topctop 16913  TopOnctopon 16914    Cn ccn 17242    tX ctx 17545   IIcii 18858   CovMap ccvm 24895
This theorem is referenced by:  cvmlift2lem9  24951
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-inf2 7552  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  ax-addf 9025  ax-mulf 9026
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-se 4502  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-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  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-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-cnp 17246  df-cmp 17404  df-con 17428  df-lly 17482  df-nlly 17483  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860  df-htpy 18948  df-phtpy 18949  df-phtpc 18970  df-pcon 24861  df-scon 24862  df-cvm 24896
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