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Theorem cvmlift2lem7 27341
Description: Lemma for cvmlift2 27348. (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 2454 . . . . . . . . 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 27337 . . . . . . . 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 1001 . . . . . 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 5800 . . . . 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 1000 . . . . . . 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 20588 . . . . . . . 8  |-  ( 0 [,] 1 )  = 
U. II
1413, 1cnf 18981 . . . . . . 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 5876 . . . . . 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 6207 . . . . . . 7  |-  ( z  =  y  ->  (
x G z )  =  ( x G y ) )
20 eqid 2454 . . . . . . 7  |-  ( z  e.  ( 0 [,] 1 )  |->  ( x G z ) )  =  ( z  e.  ( 0 [,] 1
)  |->  ( x G z ) )
21 ovex 6224 . . . . . . 7  |-  ( x G y )  e. 
_V
2219, 20, 21fvmpt 5882 . . . . . 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 2503 . . . 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 1184 . . 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 6258 . 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 5952 . . 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 27294 . . . . 5  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
31 eqid 2454 . . . . . 6  |-  U. J  =  U. J
321, 31cnf 18981 . . . . 5  |-  ( F  e.  ( C  Cn  J )  ->  F : B --> U. J )
332, 30, 323syl 20 . . . 4  |-  ( ph  ->  F : B --> U. J
)
3433feqmptd 5852 . . 3  |-  ( ph  ->  F  =  ( w  e.  B  |->  ( F `
 w ) ) )
35 fveq2 5798 . . 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 6765 . 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 20587 . . . . . 6  |-  II  e.  Top
3837, 37, 13, 13txunii 19297 . . . . 5  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  = 
U. ( II  tX  II )
3938, 31cnf 18981 . . . 4  |-  ( G  e.  ( ( II 
tX  II )  Cn  J )  ->  G : ( ( 0 [,] 1 )  X.  ( 0 [,] 1
) ) --> U. J
)
40 ffn 5666 . . . 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 6307 . . 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 2505 1  |-  ( ph  ->  ( F  o.  K
)  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   U.cuni 4198    |-> cmpt 4457    X. cxp 4945    o. ccom 4951    Fn wfn 5520   -->wf 5521   ` cfv 5525   iota_crio 6159  (class class class)co 6199    |-> cmpt2 6201   0cc0 9392   1c1 9393   [,]cicc 11413    Cn ccn 18959    tX ctx 19264   IIcii 20582   CovMap ccvm 27287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-ec 7212  df-map 7325  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-cn 18962  df-cnp 18963  df-cmp 19121  df-con 19147  df-lly 19201  df-nlly 19202  df-tx 19266  df-hmeo 19459  df-xms 20026  df-ms 20027  df-tms 20028  df-ii 20584  df-htpy 20673  df-phtpy 20674  df-phtpc 20695  df-pcon 27253  df-scon 27254  df-cvm 27288
This theorem is referenced by:  cvmlift2lem9  27343  cvmlift2lem13  27347
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