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Theorem cvmlift2lem7 23855
Description: Lemma for cvmlift2 23862. (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 2296 . . . . . . . . 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 23851 . . . . . . . 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 697 . . . . . . 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 968 . . . . . 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 5543 . . . . 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 967 . . . . . . 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 18401 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
1413, 1cnf 16992 . . . . . . 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 15 . . . . . 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 733 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
y  e.  ( 0 [,] 1 ) )
17 fvco3 5612 . . . . . 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 642 . . . . 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 5882 . . . . . . 7  |-  ( z  =  y  ->  (
x G z )  =  ( x G y ) )
20 eqid 2296 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )
21 ovex 5899 . . . . . . 7  |-  ( x G y )  e. 
_V
2219, 20, 21fvmpt 5618 . . . . . 6  |-  ( y  e.  ( 0 [,] 1 )  ->  (
( z  e.  ( 0 [,] 1 ) 
|->  ( x G z ) ) `  y
)  =  ( x G y ) )
2316, 22syl 15 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,] 1
)  /\  y  e.  ( 0 [,] 1
) ) )  -> 
( ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) ) `  y )  =  ( x G y ) )
2411, 18, 233eqtr3d 2336 . . . 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 1147 . . 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 5928 . 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 ) ) )
27 ffvelrn 5679 . . . 4  |-  ( ( ( 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
) )  ->  (
( iota_ f  e.  ( II  Cn  C ) ( ( F  o.  f )  =  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  /\  ( f `
 0 )  =  ( H `  x
) ) ) `  y )  e.  B
)
2815, 16, 27syl2anc 642 . . 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 )
29 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 ) )
3029a1i 10 . . 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 ) ) )
31 cvmcn 23808 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
32 eqid 2296 . . . . . 6  |-  U. J  =  U. J
331, 32cnf 16992 . . . . 5  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
342, 31, 333syl 18 . . . 4  |-  ( ph  ->  F : B --> U. J
)
3534feqmptd 5591 . . 3  |-  ( ph  ->  F  =  ( w  e.  B  |->  ( F `
 w ) ) )
36 fveq2 5541 . . 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 ) ) )
3728, 30, 35, 36fmpt2co 6218 . 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 ) ) ) )
38 iitop 18400 . . . . . 6  |-  II  e.  Top
3938, 38, 13, 13txunii 17304 . . . . 5  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
4039, 32cnf 16992 . . . 4  |-  ( G  e.  ( ( II 
tX  II )  Cn  J )  ->  G : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> U. J
)
41 ffn 5405 . . . 4  |-  ( G : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> U. J  ->  G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
423, 40, 413syl 18 . . 3  |-  ( ph  ->  G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )
43 fnov 5968 . . 3  |-  ( G  Fn  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) )  <->  G  =  ( x  e.  (
0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  ( x G y ) ) )
4442, 43sylib 188 . 2  |-  ( ph  ->  G  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( x G y ) ) )
4526, 37, 443eqtr4d 2338 1  |-  ( ph  ->  ( F  o.  K
)  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   U.cuni 3843    e. cmpt 4093    X. cxp 4703    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   iota_crio 6313   0cc0 8753   1c1 8754   [,]cicc 10675    Cn ccn 16970    tX ctx 17271   IIcii 18395   CovMap ccvm 23801
This theorem is referenced by:  cvmlift2lem9  23857  cvmlift2lem13  23861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-cn 16973  df-cnp 16974  df-cmp 17130  df-con 17154  df-lly 17208  df-nlly 17209  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-ii 18397  df-htpy 18484  df-phtpy 18485  df-phtpc 18506  df-pcon 23767  df-scon 23768  df-cvm 23802
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