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Theorem ptcldmpt 20564
Description: A closed box in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Hypotheses
Ref Expression
ptcldmpt.a  |-  ( ph  ->  A  e.  V )
ptcldmpt.j  |-  ( (
ph  /\  k  e.  A )  ->  J  e.  Top )
ptcldmpt.c  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  J
) )
Assertion
Ref Expression
ptcldmpt  |-  ( ph  -> 
X_ k  e.  A  C  e.  ( Clsd `  ( Xt_ `  (
k  e.  A  |->  J ) ) ) )
Distinct variable groups:    ph, k    A, k
Allowed substitution hints:    C( k)    J( k)    V( k)

Proof of Theorem ptcldmpt
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 nfcv 2591 . . 3  |-  F/_ l C
2 nfcsb1v 3417 . . 3  |-  F/_ k [_ l  /  k ]_ C
3 csbeq1a 3410 . . 3  |-  ( k  =  l  ->  C  =  [_ l  /  k ]_ C )
41, 2, 3cbvixp 7547 . 2  |-  X_ k  e.  A  C  =  X_ l  e.  A  [_ l  /  k ]_ C
5 ptcldmpt.a . . 3  |-  ( ph  ->  A  e.  V )
6 ptcldmpt.j . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  J  e.  Top )
7 eqid 2429 . . . 4  |-  ( k  e.  A  |->  J )  =  ( k  e.  A  |->  J )
86, 7fmptd 6061 . . 3  |-  ( ph  ->  ( k  e.  A  |->  J ) : A --> Top )
9 nfv 1754 . . . . 5  |-  F/ k ( ph  /\  l  e.  A )
10 nfcv 2591 . . . . . . 7  |-  F/_ k Clsd
11 nffvmpt1 5889 . . . . . . 7  |-  F/_ k
( ( k  e.  A  |->  J ) `  l )
1210, 11nffv 5888 . . . . . 6  |-  F/_ k
( Clsd `  ( (
k  e.  A  |->  J ) `  l ) )
132, 12nfel 2604 . . . . 5  |-  F/ k
[_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) )
149, 13nfim 1978 . . . 4  |-  F/ k ( ( ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) ) )
15 eleq1 2501 . . . . . 6  |-  ( k  =  l  ->  (
k  e.  A  <->  l  e.  A ) )
1615anbi2d 708 . . . . 5  |-  ( k  =  l  ->  (
( ph  /\  k  e.  A )  <->  ( ph  /\  l  e.  A ) ) )
17 fveq2 5881 . . . . . . 7  |-  ( k  =  l  ->  (
( k  e.  A  |->  J ) `  k
)  =  ( ( k  e.  A  |->  J ) `  l ) )
1817fveq2d 5885 . . . . . 6  |-  ( k  =  l  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
193, 18eleq12d 2511 . . . . 5  |-  ( k  =  l  ->  ( C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `  k ) )  <->  [_ l  / 
k ]_ C  e.  (
Clsd `  ( (
k  e.  A  |->  J ) `  l ) ) ) )
2016, 19imbi12d 321 . . . 4  |-  ( k  =  l  ->  (
( ( ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `  k ) ) )  <-> 
( ( ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) ) ) ) )
21 ptcldmpt.c . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  J
) )
22 simpr 462 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  A )
237fvmpt2 5973 . . . . . . 7  |-  ( ( k  e.  A  /\  J  e.  Top )  ->  ( ( k  e.  A  |->  J ) `  k )  =  J )
2422, 6, 23syl2anc 665 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( k  e.  A  |->  J ) `  k
)  =  J )
2524fveq2d 5885 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  J
) )
2621, 25eleqtrrd 2520 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  k
) ) )
2714, 20, 26chvar 2069 . . 3  |-  ( (
ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
285, 8, 27ptcld 20563 . 2  |-  ( ph  -> 
X_ l  e.  A  [_ l  /  k ]_ C  e.  ( Clsd `  ( Xt_ `  (
k  e.  A  |->  J ) ) ) )
294, 28syl5eqel 2521 1  |-  ( ph  -> 
X_ k  e.  A  C  e.  ( Clsd `  ( Xt_ `  (
k  e.  A  |->  J ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   [_csb 3401    |-> cmpt 4484   ` cfv 5601   X_cixp 7530   Xt_cpt 15300   Topctop 19852   Clsdccld 19966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-ixp 7531  df-en 7578  df-fin 7581  df-fi 7931  df-topgen 15305  df-pt 15306  df-top 19856  df-bases 19857  df-cld 19969
This theorem is referenced by:  ptclsg  20565  kelac1  35642
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