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Theorem ptcldmpt 17646
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 2572 . . 3  |-  F/_ l C
2 nfcsb1v 3283 . . 3  |-  F/_ k [_ l  /  k ]_ C
3 csbeq1a 3259 . . 3  |-  ( k  =  l  ->  C  =  [_ l  /  k ]_ C )
41, 2, 3cbvixp 7079 . 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 2436 . . . 4  |-  ( k  e.  A  |->  J )  =  ( k  e.  A  |->  J )
86, 7fmptd 5893 . . 3  |-  ( ph  ->  ( k  e.  A  |->  J ) : A --> Top )
9 nfv 1629 . . . . 5  |-  F/ k ( ph  /\  l  e.  A )
10 nfcv 2572 . . . . . . 7  |-  F/_ k Clsd
11 nffvmpt1 5736 . . . . . . 7  |-  F/_ k
( ( k  e.  A  |->  J ) `  l )
1210, 11nffv 5735 . . . . . 6  |-  F/_ k
( Clsd `  ( (
k  e.  A  |->  J ) `  l ) )
132, 12nfel 2580 . . . . 5  |-  F/ k
[_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) )
149, 13nfim 1832 . . . 4  |-  F/ k ( ( ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) ) )
15 eleq1 2496 . . . . . 6  |-  ( k  =  l  ->  (
k  e.  A  <->  l  e.  A ) )
1615anbi2d 685 . . . . 5  |-  ( k  =  l  ->  (
( ph  /\  k  e.  A )  <->  ( ph  /\  l  e.  A ) ) )
17 fveq2 5728 . . . . . . 7  |-  ( k  =  l  ->  (
( k  e.  A  |->  J ) `  k
)  =  ( ( k  e.  A  |->  J ) `  l ) )
1817fveq2d 5732 . . . . . 6  |-  ( k  =  l  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
193, 18eleq12d 2504 . . . . 5  |-  ( k  =  l  ->  ( C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `  k ) )  <->  [_ l  / 
k ]_ C  e.  (
Clsd `  ( (
k  e.  A  |->  J ) `  l ) ) ) )
2016, 19imbi12d 312 . . . 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 448 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  A )
237fvmpt2 5812 . . . . . . 7  |-  ( ( k  e.  A  /\  J  e.  Top )  ->  ( ( k  e.  A  |->  J ) `  k )  =  J )
2422, 6, 23syl2anc 643 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( k  e.  A  |->  J ) `  k
)  =  J )
2524fveq2d 5732 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  J
) )
2621, 25eleqtrrd 2513 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  k
) ) )
2714, 20, 26chvar 1968 . . 3  |-  ( (
ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
285, 8, 27ptcld 17645 . 2  |-  ( ph  -> 
X_ l  e.  A  [_ l  /  k ]_ C  e.  ( Clsd `  ( Xt_ `  (
k  e.  A  |->  J ) ) ) )
294, 28syl5eqel 2520 1  |-  ( ph  -> 
X_ k  e.  A  C  e.  ( Clsd `  ( Xt_ `  (
k  e.  A  |->  J ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   [_csb 3251    e. cmpt 4266   ` cfv 5454   X_cixp 7063   Xt_cpt 13666   Topctop 16958   Clsdccld 17080
This theorem is referenced by:  ptclsg  17647  kelac1  27138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-ixp 7064  df-en 7110  df-fin 7113  df-fi 7416  df-topgen 13667  df-pt 13668  df-top 16963  df-bases 16965  df-cld 17083
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