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Theorem ptcldmpt 19185
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 2577 . . 3  |-  F/_ l C
2 nfcsb1v 3302 . . 3  |-  F/_ k [_ l  /  k ]_ C
3 csbeq1a 3295 . . 3  |-  ( k  =  l  ->  C  =  [_ l  /  k ]_ C )
41, 2, 3cbvixp 7278 . 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 2441 . . . 4  |-  ( k  e.  A  |->  J )  =  ( k  e.  A  |->  J )
86, 7fmptd 5865 . . 3  |-  ( ph  ->  ( k  e.  A  |->  J ) : A --> Top )
9 nfv 1673 . . . . 5  |-  F/ k ( ph  /\  l  e.  A )
10 nfcv 2577 . . . . . . 7  |-  F/_ k Clsd
11 nffvmpt1 5697 . . . . . . 7  |-  F/_ k
( ( k  e.  A  |->  J ) `  l )
1210, 11nffv 5696 . . . . . 6  |-  F/_ k
( Clsd `  ( (
k  e.  A  |->  J ) `  l ) )
132, 12nfel 2585 . . . . 5  |-  F/ k
[_ l  /  k ]_ C  e.  ( Clsd `  ( ( k  e.  A  |->  J ) `
 l ) )
149, 13nfim 1853 . . . 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 703 . . . . 5  |-  ( k  =  l  ->  (
( ph  /\  k  e.  A )  <->  ( ph  /\  l  e.  A ) ) )
17 fveq2 5689 . . . . . . 7  |-  ( k  =  l  ->  (
( k  e.  A  |->  J ) `  k
)  =  ( ( k  e.  A  |->  J ) `  l ) )
1817fveq2d 5693 . . . . . 6  |-  ( k  =  l  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
193, 18eleq12d 2509 . . . . 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 5779 . . . . . . 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 5693 . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  ( Clsd `  ( ( k  e.  A  |->  J ) `
 k ) )  =  ( Clsd `  J
) )
2621, 25eleqtrrd 2518 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  k
) ) )
2714, 20, 26chvar 1957 . . 3  |-  ( (
ph  /\  l  e.  A )  ->  [_ l  /  k ]_ C  e.  ( Clsd `  (
( k  e.  A  |->  J ) `  l
) ) )
285, 8, 27ptcld 19184 . 2  |-  ( ph  -> 
X_ l  e.  A  [_ l  /  k ]_ C  e.  ( Clsd `  ( Xt_ `  (
k  e.  A  |->  J ) ) ) )
294, 28syl5eqel 2525 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 1369    e. wcel 1756   [_csb 3286    e. cmpt 4348   ` cfv 5416   X_cixp 7261   Xt_cpt 14375   Topctop 18496   Clsdccld 18618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-ixp 7262  df-en 7309  df-fin 7312  df-fi 7659  df-topgen 14380  df-pt 14381  df-top 18501  df-bases 18503  df-cld 18621
This theorem is referenced by:  ptclsg  19186  kelac1  29413
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