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Theorem cvmlift2lem7 28422
Description: Lemma for cvmlift2 28429. (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 2467 . . . . . . . . 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 28418 . . . . . . . 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 716 . . . . . . 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 1009 . . . . . 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 5868 . . . . 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 1008 . . . . . . 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 21148 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
1413, 1cnf 19541 . . . . . . 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 756 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
y  e.  ( 0 [,] 1 ) )
17 fvco3 5944 . . . . . 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 661 . . . . 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 6292 . . . . . . 7  |-  ( z  =  y  ->  (
x G z )  =  ( x G y ) )
20 eqid 2467 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )
21 ovex 6309 . . . . . . 7  |-  ( x G y )  e. 
_V
2219, 20, 21fvmpt 5950 . . . . . 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 2516 . . . 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 1192 . . 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 6345 . 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 6022 . . 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 28375 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
31 eqid 2467 . . . . . 6  |-  U. J  =  U. J
321, 31cnf 19541 . . . . 5  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
332, 30, 323syl 20 . . . 4  |-  ( ph  ->  F : B --> U. J
)
3433feqmptd 5920 . . 3  |-  ( ph  ->  F  =  ( w  e.  B  |->  ( F `
 w ) ) )
35 fveq2 5866 . . 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 6866 . 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 21147 . . . . . 6  |-  II  e.  Top
3837, 37, 13, 13txunii 19857 . . . . 5  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
3938, 31cnf 19541 . . . 4  |-  ( G  e.  ( ( II 
tX  II )  Cn  J )  ->  G : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> U. J
)
40 ffn 5731 . . . 4  |-  ( G : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> U. J  ->  G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
413, 39, 403syl 20 . . 3  |-  ( ph  ->  G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
42 fnov 6394 . . 3  |-  ( G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  G  =  ( x  e.  (
0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( x G y ) ) )
4341, 42sylib 196 . 2  |-  ( ph  ->  G  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( x G y ) ) )
4426, 36, 433eqtr4d 2518 1  |-  ( ph  ->  ( F  o.  K
)  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   U.cuni 4245    |-> cmpt 4505    X. cxp 4997    o. ccom 5003    Fn wfn 5583   -->wf 5584   ` cfv 5588   iota_crio 6244  (class class class)co 6284    |-> cmpt2 6286   0cc0 9492   1c1 9493   [,]cicc 11532    Cn ccn 19519    tX ctx 19824   IIcii 21142   CovMap ccvm 28368
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 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  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 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-cn 19522  df-cnp 19523  df-cmp 19681  df-con 19707  df-lly 19761  df-nlly 19762  df-tx 19826  df-hmeo 20019  df-xms 20586  df-ms 20587  df-tms 20588  df-ii 21144  df-htpy 21233  df-phtpy 21234  df-phtpc 21255  df-pcon 28334  df-scon 28335  df-cvm 28369
This theorem is referenced by:  cvmlift2lem9  28424  cvmlift2lem13  28428
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