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Theorem txdis 17656
Description: The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txdis  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX  ~P B )  =  ~P ( A  X.  B
) )

Proof of Theorem txdis
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 distop 17052 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 distop 17052 . . . . 5  |-  ( B  e.  W  ->  ~P B  e.  Top )
3 unipw 4406 . . . . . . 7  |-  U. ~P A  =  A
43eqcomi 2439 . . . . . 6  |-  A  = 
U. ~P A
5 unipw 4406 . . . . . . 7  |-  U. ~P B  =  B
65eqcomi 2439 . . . . . 6  |-  B  = 
U. ~P B
74, 6txuni 17616 . . . . 5  |-  ( ( ~P A  e.  Top  /\ 
~P B  e.  Top )  ->  ( A  X.  B )  =  U. ( ~P A  tX  ~P B ) )
81, 2, 7syl2an 464 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  =  U. ( ~P A  tX  ~P B
) )
9 eqimss2 3393 . . . 4  |-  ( ( A  X.  B )  =  U. ( ~P A  tX  ~P B
)  ->  U. ( ~P A  tX  ~P B
)  C_  ( A  X.  B ) )
108, 9syl 16 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. ( ~P A  tX 
~P B )  C_  ( A  X.  B
) )
11 sspwuni 4168 . . 3  |-  ( ( ~P A  tX  ~P B )  C_  ~P ( A  X.  B
)  <->  U. ( ~P A  tX 
~P B )  C_  ( A  X.  B
) )
1210, 11sylibr 204 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX  ~P B )  C_  ~P ( A  X.  B
) )
13 elelpwi 3801 . . . . . . . . 9  |-  ( ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) )  ->  y  e.  ( A  X.  B ) )
1413adantl 453 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  ( A  X.  B ) )
15 xp1st 6368 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 1st `  y )  e.  A )
16 snelpwi 4401 . . . . . . . 8  |-  ( ( 1st `  y )  e.  A  ->  { ( 1st `  y ) }  e.  ~P A
)
1714, 15, 163syl 19 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { ( 1st `  y ) }  e.  ~P A )
18 xp2nd 6369 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 2nd `  y )  e.  B )
19 snelpwi 4401 . . . . . . . 8  |-  ( ( 2nd `  y )  e.  B  ->  { ( 2nd `  y ) }  e.  ~P B
)
2014, 18, 193syl 19 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { ( 2nd `  y ) }  e.  ~P B )
21 vex 2951 . . . . . . . . 9  |-  y  e. 
_V
2221snid 3833 . . . . . . . 8  |-  y  e. 
{ y }
23 1st2nd2 6378 . . . . . . . . . 10  |-  ( y  e.  ( A  X.  B )  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2414, 23syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
2524sneqd 3819 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { y }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
2622, 25syl5eleq 2521 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
27 simprl 733 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  x
)
2824, 27eqeltrrd 2510 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  x
)
2928snssd 3935 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  C_  x
)
30 xpeq1 4884 . . . . . . . . . 10  |-  ( z  =  { ( 1st `  y ) }  ->  ( z  X.  w )  =  ( { ( 1st `  y ) }  X.  w ) )
3130eleq2d 2502 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( y  e.  ( z  X.  w )  <->  y  e.  ( { ( 1st `  y
) }  X.  w
) ) )
3230sseq1d 3367 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( ( z  X.  w
)  C_  x  <->  ( {
( 1st `  y
) }  X.  w
)  C_  x )
)
3331, 32anbi12d 692 . . . . . . . 8  |-  ( z  =  { ( 1st `  y ) }  ->  ( ( y  e.  ( z  X.  w )  /\  ( z  X.  w )  C_  x
)  <->  ( y  e.  ( { ( 1st `  y ) }  X.  w )  /\  ( { ( 1st `  y
) }  X.  w
)  C_  x )
) )
34 xpeq2 4885 . . . . . . . . . . 11  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } ) )
35 fvex 5734 . . . . . . . . . . . 12  |-  ( 1st `  y )  e.  _V
36 fvex 5734 . . . . . . . . . . . 12  |-  ( 2nd `  y )  e.  _V
3735, 36xpsn 5902 . . . . . . . . . . 11  |-  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } )  =  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }
3834, 37syl6eq 2483 . . . . . . . . . 10  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
3938eleq2d 2502 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( y  e.  ( { ( 1st `  y
) }  X.  w
)  <->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } ) )
4038sseq1d 3367 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( ( { ( 1st `  y ) }  X.  w )  C_  x  <->  {
<. ( 1st `  y
) ,  ( 2nd `  y ) >. }  C_  x ) )
4139, 40anbi12d 692 . . . . . . . 8  |-  ( w  =  { ( 2nd `  y ) }  ->  ( ( y  e.  ( { ( 1st `  y
) }  X.  w
)  /\  ( {
( 1st `  y
) }  X.  w
)  C_  x )  <->  ( y  e.  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  /\  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  C_  x ) ) )
4233, 41rspc2ev 3052 . . . . . . 7  |-  ( ( { ( 1st `  y
) }  e.  ~P A  /\  { ( 2nd `  y ) }  e.  ~P B  /\  (
y  e.  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  /\  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }  C_  x ) )  ->  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )
4317, 20, 26, 29, 42syl112anc 1188 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )
4443expr 599 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  y  e.  x )  ->  (
x  e.  ~P ( A  X.  B )  ->  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) ) )
4544ralrimdva 2788 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  ~P ( A  X.  B
)  ->  A. y  e.  x  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) ) )
46 eltx 17592 . . . . 5  |-  ( ( ~P A  e.  Top  /\ 
~P B  e.  Top )  ->  ( x  e.  ( ~P A  tX  ~P B )  <->  A. y  e.  x  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) ) )
471, 2, 46syl2an 464 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  ( ~P A  tX  ~P B )  <->  A. y  e.  x  E. z  e.  ~P  A E. w  e.  ~P  B ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) ) )
4845, 47sylibrd 226 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  ~P ( A  X.  B
)  ->  x  e.  ( ~P A  tX  ~P B ) ) )
4948ssrdv 3346 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  X.  B )  C_  ( ~P A  tX  ~P B
) )
5012, 49eqssd 3357 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX  ~P B )  =  ~P ( A  X.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   ~Pcpw 3791   {csn 3806   <.cop 3809   U.cuni 4007    X. cxp 4868   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   Topctop 16950    tX ctx 17584
This theorem is referenced by:  distgp  18121
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-tx 17586
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