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Theorem cvmlift2lem6 28409
Description: Lemma for cvmlift2 28417. (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 28408 . . . . . . 7  |-  ( ph  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) --> B )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  K : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> B )
10 ffn 5730 . . . . . 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 6393 . . . . 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 5271 . . 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 461 . . . . . 6  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  X  e.  ( 0 [,] 1
) )
1615snssd 4172 . . . . 5  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  { X }  C_  ( 0 [,] 1 ) )
17 ssid 3523 . . . . 5  |-  ( 0 [,] 1 )  C_  ( 0 [,] 1
)
18 resmpt2 6383 . . . . 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 662 . . . 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 4052 . . . . . . . 8  |-  ( u  e.  { X }  ->  u  =  X )
21203ad2ant2 1018 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  u  =  X )
2221oveq1d 6298 . . . . . 6  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  ( u K v )  =  ( X K v ) )
23 simp1r 1021 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  X  e.  ( 0 [,] 1 ) )
24 simp3 998 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 0 [,] 1
) )  /\  u  e.  { X }  /\  v  e.  ( 0 [,] 1 ) )  ->  v  e.  ( 0 [,] 1 ) )
251, 2, 3, 4, 5, 6, 7cvmlift2lem4 28407 . . . . . . 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 661 . . . . . 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 2508 . . . . 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 6344 . . . 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 2508 . . 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 2508 . 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 2467 . . . 4  |-  ( IIt  { X } )  =  ( IIt 
{ X } )
32 iitopon 21134 . . . . 5  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
3332a1i 11 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  II  e.  (TopOn `  ( 0 [,] 1 ) ) )
34 eqid 2467 . . . 4  |-  ( IIt  ( 0 [,] 1 ) )  =  ( IIt  ( 0 [,] 1 ) )
3517a1i 11 . . . 4  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
0 [,] 1 ) 
C_  ( 0 [,] 1 ) )
3633, 33cnmpt2nd 19921 . . . . 5  |-  ( (
ph  /\  X  e.  ( 0 [,] 1
) )  ->  (
u  e.  ( 0 [,] 1 ) ,  v  e.  ( 0 [,] 1 )  |->  v )  e.  ( ( II  tX  II )  Cn  II ) )
37 eqid 2467 . . . . . . 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 28406 . . . . . 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 1008 . . . . 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 19924 . . . 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 19929 . . 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 21135 . . . . 5  |-  II  e.  Top
43 snex 4688 . . . . 5  |-  { X }  e.  _V
44 ovex 6308 . . . . 5  |-  ( 0 [,] 1 )  e. 
_V
45 txrest 19883 . . . . 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 673 . . . 4  |-  ( ( II  tX  II )t  ( { X }  X.  (
0 [,] 1 ) ) )  =  ( ( IIt  { X } ) 
tX  ( IIt  ( 0 [,] 1 ) ) )
4746oveq1i 6293 . . 3  |-  ( ( ( II  tX  II )t  ( { X }  X.  ( 0 [,] 1
) ) )  Cn  C )  =  ( ( ( IIt  { X } )  tX  (
IIt 
( 0 [,] 1
) ) )  Cn  C )
4841, 47syl6eleqr 2566 . 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 2555 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   {csn 4027   U.cuni 4245    |-> cmpt 4505    X. cxp 4997    |` cres 5001    o. ccom 5003    Fn wfn 5582   -->wf 5583   ` cfv 5587   iota_crio 6243  (class class class)co 6283    |-> cmpt2 6285   0cc0 9491   1c1 9492   [,]cicc 11531   ↾t crest 14675   Topctop 19177  TopOnctopon 19178    Cn ccn 19507    tX ctx 19812   IIcii 21130   CovMap ccvm 28356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-sum 13471  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-cn 19510  df-cnp 19511  df-cmp 19669  df-con 19695  df-lly 19749  df-nlly 19750  df-tx 19814  df-hmeo 20007  df-xms 20574  df-ms 20575  df-tms 20576  df-ii 21132  df-htpy 21221  df-phtpy 21222  df-phtpc 21243  df-pcon 28322  df-scon 28323  df-cvm 28357
This theorem is referenced by:  cvmlift2lem9  28412
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