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Theorem cvmlift2lem7 25031
Description: Lemma for cvmlift2 25038. (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
cvmlift2lem7  |-  ( ph  ->  ( F  o.  K
)  =  G )
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    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 cvmlift2lem7
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 cvmlift2.b . . . . . . . . 9  |-  B  = 
U. C
2 cvmlift2.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
3 cvmlift2.g . . . . . . . . 9  |-  ( ph  ->  G  e.  ( ( II  tX  II )  Cn  J ) )
4 cvmlift2.p . . . . . . . . 9  |-  ( ph  ->  P  e.  B )
5 cvmlift2.i . . . . . . . . 9  |-  ( ph  ->  ( F `  P
)  =  ( 0 G 0 ) )
6 cvmlift2.h . . . . . . . . 9  |-  H  =  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( z G 0 ) )  /\  ( f `
 0 )  =  P ) )
7 eqid 2443 . . . . . . . . 9  |-  ( 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 ) ) )
81, 2, 3, 4, 5, 6, 7cvmlift2lem3 25027 . . . . . . . 8  |-  ( (
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
) ) )
98adantrr 699 . . . . . . 7  |-  ( (
ph  /\  ( 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 ) ) )  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 ) ) )
109simp2d 971 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( 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 ) ) )
1110fveq1d 5765 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( F  o.  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) ) `
 y )  =  ( ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) ) `  y ) )
129simp1d 970 . . . . . . 7  |-  ( (
ph  /\  ( 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
) ) )  e.  ( II  Cn  C
) )
13 iiuni 18949 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
1413, 1cnf 17348 . . . . . . 7  |-  ( (
iota_ f  e.  (
II  Cn  C )
( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) )  e.  ( II  Cn  C
)  ->  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )  /\  ( f `  0
)  =  ( H `
 x ) ) ) : ( 0 [,] 1 ) --> B )
1512, 14syl 16 . . . . . 6  |-  ( (
ph  /\  ( 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
) ) ) : ( 0 [,] 1
) --> B )
16 simprr 735 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
y  e.  ( 0 [,] 1 ) )
17 fvco3 5836 . . . . . 6  |-  ( ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) : ( 0 [,] 1
) --> B  /\  y  e.  ( 0 [,] 1
) )  ->  (
( F  o.  ( iota_ f  e.  ( II 
Cn  C ) ( ( F  o.  f
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f ` 
0 )  =  ( H `  x ) ) ) ) `  y )  =  ( F `  ( (
iota_ f  e.  (
II  Cn  C )
( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) `  y ) ) )
1815, 16, 17syl2anc 644 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( F  o.  ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) ) `
 y )  =  ( F `  (
( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) `  y ) ) )
19 oveq2 6125 . . . . . . 7  |-  ( z  =  y  ->  (
x G z )  =  ( x G y ) )
20 eqid 2443 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )
21 ovex 6142 . . . . . . 7  |-  ( x G y )  e. 
_V
2219, 20, 21fvmpt 5842 . . . . . 6  |-  ( y  e.  ( 0 [,] 1 )  ->  (
( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) ) `  y
)  =  ( x G y ) )
2316, 22syl 16 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) ) `  y )  =  ( x G y ) )
2411, 18, 233eqtr3d 2483 . . . 4  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( F `  (
( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) `  y ) )  =  ( x G y ) )
25243impb 1150 . . 3  |-  ( (
ph  /\  x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) )  ->  ( F `  ( ( iota_ f  e.  ( II 
Cn  C ) ( ( F  o.  f
)  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f ` 
0 )  =  ( H `  x ) ) ) `  y
) )  =  ( x G y ) )
2625mpt2eq3dva 6174 . 2  |-  ( ph  ->  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( F `  (
( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) `  y ) ) )  =  ( x  e.  ( 0 [,] 1
) ,  y  e.  ( 0 [,] 1
)  |->  ( x G y ) ) )
2715, 16ffvelrnd 5907 . . 3  |-  ( (
ph  /\  ( 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 )  e.  B )
28 cvmlift2.k . . . 4  |-  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 ) )
2928a1i 11 . . 3  |-  ( ph  ->  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 ) ) )
30 cvmcn 24984 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
31 eqid 2443 . . . . . 6  |-  U. J  =  U. J
321, 31cnf 17348 . . . . 5  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
332, 30, 323syl 19 . . . 4  |-  ( ph  ->  F : B --> U. J
)
3433feqmptd 5815 . . 3  |-  ( ph  ->  F  =  ( w  e.  B  |->  ( F `
 w ) ) )
35 fveq2 5763 . . 3  |-  ( w  =  ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )  /\  ( f `  0
)  =  ( H `
 x ) ) ) `  y )  ->  ( F `  w )  =  ( F `  ( (
iota_ f  e.  (
II  Cn  C )
( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) `  y ) ) )
3627, 29, 34, 35fmpt2co 6466 . 2  |-  ( ph  ->  ( F  o.  K
)  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( F `
 ( ( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )  /\  ( f `  0
)  =  ( H `
 x ) ) ) `  y ) ) ) )
37 iitop 18948 . . . . . 6  |-  II  e.  Top
3837, 37, 13, 13txunii 17663 . . . . 5  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
3938, 31cnf 17348 . . . 4  |-  ( G  e.  ( ( II 
tX  II )  Cn  J )  ->  G : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> U. J
)
40 ffn 5626 . . . 4  |-  ( G : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> U. J  ->  G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
413, 39, 403syl 19 . . 3  |-  ( ph  ->  G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
42 fnov 6214 . . 3  |-  ( G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  G  =  ( x  e.  (
0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( x G y ) ) )
4341, 42sylib 190 . 2  |-  ( ph  ->  G  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( x G y ) ) )
4426, 36, 433eqtr4d 2485 1  |-  ( ph  ->  ( F  o.  K
)  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   U.cuni 4044    e. cmpt 4297    X. cxp 4911    o. ccom 4917    Fn wfn 5484   -->wf 5485   ` cfv 5489  (class class class)co 6117    e. cmpt2 6119   iota_crio 6578   0cc0 9028   1c1 9029   [,]cicc 10957    Cn ccn 17326    tX ctx 17630   IIcii 18943   CovMap ccvm 24977
This theorem is referenced by:  cvmlift2lem9  25033  cvmlift2lem13  25037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-inf2 7632  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105  ax-pre-sup 9106  ax-addf 9107  ax-mulf 9108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-iin 4125  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-se 4577  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-isom 5498  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-of 6341  df-1st 6385  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-1o 6760  df-2o 6761  df-oadd 6764  df-er 6941  df-ec 6943  df-map 7056  df-ixp 7100  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-fi 7452  df-sup 7482  df-oi 7515  df-card 7864  df-cda 8086  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-div 9716  df-nn 10039  df-2 10096  df-3 10097  df-4 10098  df-5 10099  df-6 10100  df-7 10101  df-8 10102  df-9 10103  df-10 10104  df-n0 10260  df-z 10321  df-dec 10421  df-uz 10527  df-q 10613  df-rp 10651  df-xneg 10748  df-xadd 10749  df-xmul 10750  df-ioo 10958  df-ico 10960  df-icc 10961  df-fz 11082  df-fzo 11174  df-fl 11240  df-seq 11362  df-exp 11421  df-hash 11657  df-cj 11942  df-re 11943  df-im 11944  df-sqr 12078  df-abs 12079  df-clim 12320  df-sum 12518  df-struct 13509  df-ndx 13510  df-slot 13511  df-base 13512  df-sets 13513  df-ress 13514  df-plusg 13580  df-mulr 13581  df-starv 13582  df-sca 13583  df-vsca 13584  df-tset 13586  df-ple 13587  df-ds 13589  df-unif 13590  df-hom 13591  df-cco 13592  df-rest 13688  df-topn 13689  df-topgen 13705  df-pt 13706  df-prds 13709  df-xrs 13764  df-0g 13765  df-gsum 13766  df-qtop 13771  df-imas 13772  df-xps 13774  df-mre 13849  df-mrc 13850  df-acs 13852  df-mnd 14728  df-submnd 14777  df-mulg 14853  df-cntz 15154  df-cmn 15452  df-psmet 16732  df-xmet 16733  df-met 16734  df-bl 16735  df-mopn 16736  df-cnfld 16742  df-top 17001  df-bases 17003  df-topon 17004  df-topsp 17005  df-cld 17121  df-ntr 17122  df-cls 17123  df-nei 17200  df-cn 17329  df-cnp 17330  df-cmp 17488  df-con 17513  df-lly 17567  df-nlly 17568  df-tx 17632  df-hmeo 17825  df-xms 18388  df-ms 18389  df-tms 18390  df-ii 18945  df-htpy 19033  df-phtpy 19034  df-phtpc 19055  df-pcon 24943  df-scon 24944  df-cvm 24978
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