MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  txdis Unicode version

Theorem txdis 17342
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 16749 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
2 distop 16749 . . . . 5  |-  ( B  e.  W  ->  ~P B  e.  Top )
3 unipw 4240 . . . . . . 7  |-  U. ~P A  =  A
43eqcomi 2300 . . . . . 6  |-  A  = 
U. ~P A
5 unipw 4240 . . . . . . 7  |-  U. ~P B  =  B
65eqcomi 2300 . . . . . 6  |-  B  = 
U. ~P B
74, 6txuni 17303 . . . . 5  |-  ( ( ~P A  e.  Top  /\ 
~P B  e.  Top )  ->  ( A  X.  B )  =  U. ( ~P A  tX  ~P B ) )
81, 2, 7syl2an 463 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  X.  B
)  =  U. ( ~P A  tX  ~P B
) )
9 eqimss2 3244 . . . 4  |-  ( ( A  X.  B )  =  U. ( ~P A  tX  ~P B
)  ->  U. ( ~P A  tX  ~P B
)  C_  ( A  X.  B ) )
108, 9syl 15 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. ( ~P A  tX 
~P B )  C_  ( A  X.  B
) )
11 sspwuni 4003 . . 3  |-  ( ( ~P A  tX  ~P B )  C_  ~P ( A  X.  B
)  <->  U. ( ~P A  tX 
~P B )  C_  ( A  X.  B
) )
1210, 11sylibr 203 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  tX  ~P B )  C_  ~P ( A  X.  B
) )
13 elelpwi 3648 . . . . . . . . 9  |-  ( ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) )  ->  y  e.  ( A  X.  B ) )
1413adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  ( A  X.  B ) )
15 xp1st 6165 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 1st `  y )  e.  A )
16 snelpwi 4236 . . . . . . . 8  |-  ( ( 1st `  y )  e.  A  ->  { ( 1st `  y ) }  e.  ~P A
)
1714, 15, 163syl 18 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { ( 1st `  y ) }  e.  ~P A )
18 xp2nd 6166 . . . . . . . 8  |-  ( y  e.  ( A  X.  B )  ->  ( 2nd `  y )  e.  B )
19 snelpwi 4236 . . . . . . . 8  |-  ( ( 2nd `  y )  e.  B  ->  { ( 2nd `  y ) }  e.  ~P B
)
2014, 18, 193syl 18 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { ( 2nd `  y ) }  e.  ~P B )
21 vex 2804 . . . . . . . . 9  |-  y  e. 
_V
2221snid 3680 . . . . . . . 8  |-  y  e. 
{ y }
23 1st2nd2 6175 . . . . . . . . . 10  |-  ( y  e.  ( A  X.  B )  ->  y  =  <. ( 1st `  y
) ,  ( 2nd `  y ) >. )
2414, 23syl 15 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
2524sneqd 3666 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { y }  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
2622, 25syl5eleq 2382 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } )
27 simprl 732 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  y  e.  x
)
2824, 27eqeltrrd 2371 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  x
)
2928snssd 3776 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( y  e.  x  /\  x  e.  ~P ( A  X.  B ) ) )  ->  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. }  C_  x
)
30 xpeq1 4719 . . . . . . . . . 10  |-  ( z  =  { ( 1st `  y ) }  ->  ( z  X.  w )  =  ( { ( 1st `  y ) }  X.  w ) )
3130eleq2d 2363 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( y  e.  ( z  X.  w )  <->  y  e.  ( { ( 1st `  y
) }  X.  w
) ) )
3230sseq1d 3218 . . . . . . . . 9  |-  ( z  =  { ( 1st `  y ) }  ->  ( ( z  X.  w
)  C_  x  <->  ( {
( 1st `  y
) }  X.  w
)  C_  x )
)
3331, 32anbi12d 691 . . . . . . . 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 4720 . . . . . . . . . . 11  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } ) )
35 fvex 5555 . . . . . . . . . . . 12  |-  ( 1st `  y )  e.  _V
36 fvex 5555 . . . . . . . . . . . 12  |-  ( 2nd `  y )  e.  _V
3735, 36xpsn 5716 . . . . . . . . . . 11  |-  ( { ( 1st `  y
) }  X.  {
( 2nd `  y
) } )  =  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. }
3834, 37syl6eq 2344 . . . . . . . . . 10  |-  ( w  =  { ( 2nd `  y ) }  ->  ( { ( 1st `  y
) }  X.  w
)  =  { <. ( 1st `  y ) ,  ( 2nd `  y
) >. } )
3938eleq2d 2363 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( y  e.  ( { ( 1st `  y
) }  X.  w
)  <->  y  e.  { <. ( 1st `  y
) ,  ( 2nd `  y ) >. } ) )
4038sseq1d 3218 . . . . . . . . 9  |-  ( w  =  { ( 2nd `  y ) }  ->  ( ( { ( 1st `  y ) }  X.  w )  C_  x  <->  {
<. ( 1st `  y
) ,  ( 2nd `  y ) >. }  C_  x ) )
4139, 40anbi12d 691 . . . . . . . 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 2905 . . . . . . 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 1186 . . . . . 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 598 . . . . 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 2646 . . . 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 17279 . . . . 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 463 . . . 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 225 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( x  e.  ~P ( A  X.  B
)  ->  x  e.  ( ~P A  tX  ~P B ) ) )
4948ssrdv 3198 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  X.  B )  C_  ( ~P A  tX  ~P B
) )
5012, 49eqssd 3209 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   {csn 3653   <.cop 3656   U.cuni 3843    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Topctop 16647    tX ctx 17271
This theorem is referenced by:  distgp  17798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-tx 17273
  Copyright terms: Public domain W3C validator