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Theorem ptcldmpt 19878
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 2629 . . 3  |-  F/_ l C
2 nfcsb1v 3451 . . 3  |-  F/_ k [_ l  /  k ]_ C
3 csbeq1a 3444 . . 3  |-  ( k  =  l  ->  C  =  [_ l  /  k ]_ C )
41, 2, 3cbvixp 7486 . 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 2467 . . . 4  |-  ( k  e.  A  |->  J )  =  ( k  e.  A  |->  J )
86, 7fmptd 6045 . . 3  |-  ( ph  ->  ( k  e.  A  |->  J ) : A --> Top )
9 nfv 1683 . . . . 5  |-  F/ k ( ph  /\  l  e.  A )
10 nfcv 2629 . . . . . . 7  |-  F/_ k Clsd
11 nffvmpt1 5874 . . . . . . 7  |-  F/_ k
( ( k  e.  A  |->  J ) `  l )
1210, 11nffv 5873 . . . . . 6  |-  F/_ k
( Clsd `  ( (
k  e.  A  |->  J ) `  l ) )
132, 12nfel 2642 . . . . 5  |-  F/ k
[_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) )
149, 13nfim 1867 . . . 4  |-  F/ k ( ( ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) ) )
15 eleq1 2539 . . . . . 6  |-  ( k  =  l  ->  (
k  e.  A  <->  l  e.  A ) )
1615anbi2d 703 . . . . 5  |-  ( k  =  l  ->  (
( ph  /\  k  e.  A )  <->  ( ph  /\  l  e.  A ) ) )
17 fveq2 5866 . . . . . . 7  |-  ( k  =  l  ->  (
( k  e.  A  |->  J ) `  k
)  =  ( ( k  e.  A  |->  J ) `  l ) )
1817fveq2d 5870 . . . . . 6  |-  ( k  =  l  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
193, 18eleq12d 2549 . . . . 5  |-  ( k  =  l  ->  ( C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `  k ) )  <->  [_ l  / 
k ]_ C  e.  (
Clsd `  ( (
k  e.  A  |->  J ) `  l ) ) ) )
2016, 19imbi12d 320 . . . 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 461 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  A )
237fvmpt2 5957 . . . . . . 7  |-  ( ( k  e.  A  /\  J  e.  Top )  ->  ( ( k  e.  A  |->  J ) `  k )  =  J )
2422, 6, 23syl2anc 661 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( k  e.  A  |->  J ) `  k
)  =  J )
2524fveq2d 5870 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  J
) )
2621, 25eleqtrrd 2558 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  k
) ) )
2714, 20, 26chvar 1982 . . 3  |-  ( (
ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
285, 8, 27ptcld 19877 . 2  |-  ( ph  -> 
X_ l  e.  A  [_ l  /  k ]_ C  e.  ( Clsd `  ( Xt_ `  (
k  e.  A  |->  J ) ) ) )
294, 28syl5eqel 2559 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 369    = wceq 1379    e. wcel 1767   [_csb 3435    |-> cmpt 4505   ` cfv 5588   X_cixp 7469   Xt_cpt 14694   Topctop 19189   Clsdccld 19311
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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-ral 2819  df-rex 2820  df-reu 2821  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-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-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-ixp 7470  df-en 7517  df-fin 7520  df-fi 7871  df-topgen 14699  df-pt 14700  df-top 19194  df-bases 19196  df-cld 19314
This theorem is referenced by:  ptclsg  19879  kelac1  30641
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