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Theorem ptcldmpt 17308
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 2419 . . 3  |-  F/_ l C
2 nfcsb1v 3113 . . 3  |-  F/_ k [_ l  /  k ]_ C
3 csbeq1a 3089 . . 3  |-  ( k  =  l  ->  C  =  [_ l  /  k ]_ C )
41, 2, 3cbvixp 6833 . 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 2283 . . . 4  |-  ( k  e.  A  |->  J )  =  ( k  e.  A  |->  J )
86, 7fmptd 5684 . . 3  |-  ( ph  ->  ( k  e.  A  |->  J ) : A --> Top )
9 nfv 1605 . . . . 5  |-  F/ k ( ph  /\  l  e.  A )
10 nfcv 2419 . . . . . . 7  |-  F/_ k Clsd
11 nfmpt1 4109 . . . . . . . 8  |-  F/_ k
( k  e.  A  |->  J )
12 nfcv 2419 . . . . . . . 8  |-  F/_ k
l
1311, 12nffv 5532 . . . . . . 7  |-  F/_ k
( ( k  e.  A  |->  J ) `  l )
1410, 13nffv 5532 . . . . . 6  |-  F/_ k
( Clsd `  ( (
k  e.  A  |->  J ) `  l ) )
152, 14nfel 2427 . . . . 5  |-  F/ k
[_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) )
169, 15nfim 1769 . . . 4  |-  F/ k ( ( ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) ) )
17 eleq1 2343 . . . . . 6  |-  ( k  =  l  ->  (
k  e.  A  <->  l  e.  A ) )
1817anbi2d 684 . . . . 5  |-  ( k  =  l  ->  (
( ph  /\  k  e.  A )  <->  ( ph  /\  l  e.  A ) ) )
19 fveq2 5525 . . . . . . 7  |-  ( k  =  l  ->  (
( k  e.  A  |->  J ) `  k
)  =  ( ( k  e.  A  |->  J ) `  l ) )
2019fveq2d 5529 . . . . . 6  |-  ( k  =  l  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
213, 20eleq12d 2351 . . . . 5  |-  ( k  =  l  ->  ( C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `  k ) )  <->  [_ l  / 
k ]_ C  e.  (
Clsd `  ( (
k  e.  A  |->  J ) `  l ) ) ) )
2218, 21imbi12d 311 . . . 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 ) ) ) ) )
23 ptcldmpt.c . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  J
) )
24 simpr 447 . . . . . . 7  |-  ( (
ph  /\  k  e.  A )  ->  k  e.  A )
257fvmpt2 5608 . . . . . . 7  |-  ( ( k  e.  A  /\  J  e.  Top )  ->  ( ( k  e.  A  |->  J ) `  k )  =  J )
2624, 6, 25syl2anc 642 . . . . . 6  |-  ( (
ph  /\  k  e.  A )  ->  (
( k  e.  A  |->  J ) `  k
)  =  J )
2726fveq2d 5529 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  J
) )
2823, 27eleqtrrd 2360 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  k
) ) )
2916, 22, 28chvar 1926 . . 3  |-  ( (
ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
305, 8, 29ptcld 17307 . 2  |-  ( ph  -> 
X_ l  e.  A  [_ l  /  k ]_ C  e.  ( Clsd `  ( Xt_ `  (
k  e.  A  |->  J ) ) ) )
314, 30syl5eqel 2367 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 358    = wceq 1623    e. wcel 1684   [_csb 3081    e. cmpt 4077   ` cfv 5255   X_cixp 6817   Xt_cpt 13343   Topctop 16631   Clsdccld 16753
This theorem is referenced by:  ptclsg  17309  kelac1  27161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ixp 6818  df-en 6864  df-fin 6867  df-fi 7165  df-topgen 13344  df-pt 13345  df-top 16636  df-bases 16638  df-cld 16756
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