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Theorem txindis 17328
Description: The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txindis  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( A  X.  B
) }

Proof of Theorem txindis
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neq0 3465 . . . . . . 7  |-  ( -.  x  =  (/)  <->  E. y 
y  e.  x )
2 indistop 16739 . . . . . . . . . . 11  |-  { (/) ,  A }  e.  Top
3 indistop 16739 . . . . . . . . . . 11  |-  { (/) ,  B }  e.  Top
4 eltx 17263 . . . . . . . . . . 11  |-  ( ( { (/) ,  A }  e.  Top  /\  { (/) ,  B }  e.  Top )  ->  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  <->  A. y  e.  x  E. z  e.  { (/) ,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) ) )
52, 3, 4mp2an 653 . . . . . . . . . 10  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  <->  A. y  e.  x  E. z  e.  { (/) ,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x ) )
6 rsp 2603 . . . . . . . . . 10  |-  ( A. y  e.  x  E. z  e.  { (/) ,  A } E. w  e.  { (/)
,  B }  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )  ->  ( y  e.  x  ->  E. z  e.  { (/)
,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) ) )
75, 6sylbi 187 . . . . . . . . 9  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( y  e.  x  ->  E. z  e.  { (/)
,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) ) )
8 elssuni 3855 . . . . . . . . . . . . . 14  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  C_  U. ( {
(/) ,  A }  tX  { (/) ,  B }
) )
9 indisuni 16740 . . . . . . . . . . . . . . 15  |-  (  _I 
`  A )  = 
U. { (/) ,  A }
10 indisuni 16740 . . . . . . . . . . . . . . 15  |-  (  _I 
`  B )  = 
U. { (/) ,  B }
112, 3, 9, 10txunii 17288 . . . . . . . . . . . . . 14  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  = 
U. ( { (/) ,  A }  tX  { (/) ,  B } )
128, 11syl6sseqr 3225 . . . . . . . . . . . . 13  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  C_  ( (  _I 
`  A )  X.  (  _I  `  B
) ) )
1312ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( z  e.  { (/) ,  A }  /\  w  e.  { (/) ,  B } ) )  /\  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )  ->  x  C_  ( (  _I  `  A )  X.  (  _I  `  B ) ) )
14 ne0i 3461 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  ( z  X.  w )  ->  (
z  X.  w )  =/=  (/) )
1514ad2antrl 708 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  =/=  (/) )
16 xpnz 5099 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  =/=  (/)  /\  w  =/=  (/) )  <->  ( z  X.  w )  =/=  (/) )
1715, 16sylibr 203 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  =/=  (/)  /\  w  =/=  (/) ) )
1817simpld 445 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  =/=  (/) )
1918neneqd 2462 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  -.  z  =  (/) )
20 simpll 730 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  e.  {
(/) ,  A }
)
21 indislem 16737 . . . . . . . . . . . . . . . . . . 19  |-  { (/) ,  (  _I  `  A
) }  =  { (/)
,  A }
2220, 21syl6eleqr 2374 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  e.  {
(/) ,  (  _I  `  A ) } )
23 elpri 3660 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  { (/) ,  (  _I  `  A ) }  ->  ( z  =  (/)  \/  z  =  (  _I  `  A
) ) )
2422, 23syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  =  (/)  \/  z  =  (  _I  `  A
) ) )
2524ord 366 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( -.  z  =  (/)  ->  z  =  (  _I  `  A
) ) )
2619, 25mpd 14 . . . . . . . . . . . . . . 15  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  z  =  (  _I  `  A ) )
2717simprd 449 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  =/=  (/) )
2827neneqd 2462 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  -.  w  =  (/) )
29 simplr 731 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  e.  {
(/) ,  B }
)
30 indislem 16737 . . . . . . . . . . . . . . . . . . 19  |-  { (/) ,  (  _I  `  B
) }  =  { (/)
,  B }
3129, 30syl6eleqr 2374 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  e.  {
(/) ,  (  _I  `  B ) } )
32 elpri 3660 . . . . . . . . . . . . . . . . . 18  |-  ( w  e.  { (/) ,  (  _I  `  B ) }  ->  ( w  =  (/)  \/  w  =  (  _I  `  B
) ) )
3331, 32syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( w  =  (/)  \/  w  =  (  _I  `  B
) ) )
3433ord 366 . . . . . . . . . . . . . . . 16  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( -.  w  =  (/)  ->  w  =  (  _I  `  B
) ) )
3528, 34mpd 14 . . . . . . . . . . . . . . 15  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  w  =  (  _I  `  B ) )
3626, 35xpeq12d 4714 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) )
37 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( z  X.  w )  C_  x
)
3836, 37eqsstr3d 3213 . . . . . . . . . . . . 13  |-  ( ( ( z  e.  { (/)
,  A }  /\  w  e.  { (/) ,  B } )  /\  (
y  e.  ( z  X.  w )  /\  ( z  X.  w
)  C_  x )
)  ->  ( (  _I  `  A )  X.  (  _I  `  B
) )  C_  x
)
3938adantll 694 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( z  e.  { (/) ,  A }  /\  w  e.  { (/) ,  B } ) )  /\  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )  ->  (
(  _I  `  A
)  X.  (  _I 
`  B ) ) 
C_  x )
4013, 39eqssd 3196 . . . . . . . . . . 11  |-  ( ( ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( z  e.  { (/) ,  A }  /\  w  e.  { (/) ,  B } ) )  /\  ( y  e.  ( z  X.  w
)  /\  ( z  X.  w )  C_  x
) )  ->  x  =  ( (  _I 
`  A )  X.  (  _I  `  B
) ) )
4140ex 423 . . . . . . . . . 10  |-  ( ( x  e.  ( {
(/) ,  A }  tX  { (/) ,  B }
)  /\  ( z  e.  { (/) ,  A }  /\  w  e.  { (/) ,  B } ) )  ->  ( ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x )  ->  x  =  ( (  _I  `  A )  X.  (  _I  `  B
) ) ) )
4241rexlimdvva 2674 . . . . . . . . 9  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( E. z  e. 
{ (/) ,  A } E. w  e.  { (/) ,  B }  ( y  e.  ( z  X.  w )  /\  (
z  X.  w ) 
C_  x )  ->  x  =  ( (  _I  `  A )  X.  (  _I  `  B
) ) ) )
437, 42syld 40 . . . . . . . 8  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( y  e.  x  ->  x  =  ( (  _I  `  A )  X.  (  _I  `  B ) ) ) )
4443exlimdv 1664 . . . . . . 7  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( E. y  y  e.  x  ->  x  =  ( (  _I 
`  A )  X.  (  _I  `  B
) ) ) )
451, 44syl5bi 208 . . . . . 6  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( -.  x  =  (/)  ->  x  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
4645orrd 367 . . . . 5  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  -> 
( x  =  (/)  \/  x  =  ( (  _I  `  A )  X.  (  _I  `  B ) ) ) )
47 vex 2791 . . . . . 6  |-  x  e. 
_V
4847elpr 3658 . . . . 5  |-  ( x  e.  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) }  <->  ( x  =  (/)  \/  x  =  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
4946, 48sylibr 203 . . . 4  |-  ( x  e.  ( { (/) ,  A }  tX  { (/) ,  B } )  ->  x  e.  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) } )
5049ssriv 3184 . . 3  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  C_  { (/) ,  ( (  _I  `  A
)  X.  (  _I 
`  B ) ) }
519toptopon 16671 . . . . . . 7  |-  ( {
(/) ,  A }  e.  Top  <->  { (/) ,  A }  e.  (TopOn `  (  _I  `  A ) ) )
522, 51mpbi 199 . . . . . 6  |-  { (/) ,  A }  e.  (TopOn `  (  _I  `  A
) )
5310toptopon 16671 . . . . . . 7  |-  ( {
(/) ,  B }  e.  Top  <->  { (/) ,  B }  e.  (TopOn `  (  _I  `  B ) ) )
543, 53mpbi 199 . . . . . 6  |-  { (/) ,  B }  e.  (TopOn `  (  _I  `  B
) )
55 txtopon 17286 . . . . . 6  |-  ( ( { (/) ,  A }  e.  (TopOn `  (  _I  `  A ) )  /\  {
(/) ,  B }  e.  (TopOn `  (  _I  `  B ) ) )  ->  ( { (/) ,  A }  tX  { (/) ,  B } )  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B
) ) ) )
5652, 54, 55mp2an 653 . . . . 5  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B ) ) )
57 topgele 16672 . . . . 5  |-  ( ( { (/) ,  A }  tX  { (/) ,  B }
)  e.  (TopOn `  ( (  _I  `  A )  X.  (  _I  `  B ) ) )  ->  ( { (/)
,  ( (  _I 
`  A )  X.  (  _I  `  B
) ) }  C_  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( { (/)
,  A }  tX  {
(/) ,  B }
)  C_  ~P (
(  _I  `  A
)  X.  (  _I 
`  B ) ) ) )
5856, 57ax-mp 8 . . . 4  |-  ( {
(/) ,  ( (  _I  `  A )  X.  (  _I  `  B
) ) }  C_  ( { (/) ,  A }  tX  { (/) ,  B }
)  /\  ( { (/)
,  A }  tX  {
(/) ,  B }
)  C_  ~P (
(  _I  `  A
)  X.  (  _I 
`  B ) ) )
5958simpli 444 . . 3  |-  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }  C_  ( { (/)
,  A }  tX  {
(/) ,  B }
)
6050, 59eqssi 3195 . 2  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }
61 txindislem 17327 . . 3  |-  ( (  _I  `  A )  X.  (  _I  `  B ) )  =  (  _I  `  ( A  X.  B ) )
6261preq2i 3710 . 2  |-  { (/) ,  ( (  _I  `  A )  X.  (  _I  `  B ) ) }  =  { (/) ,  (  _I  `  ( A  X.  B ) ) }
63 indislem 16737 . 2  |-  { (/) ,  (  _I  `  ( A  X.  B ) ) }  =  { (/) ,  ( A  X.  B
) }
6460, 62, 633eqtri 2307 1  |-  ( {
(/) ,  A }  tX  { (/) ,  B }
)  =  { (/) ,  ( A  X.  B
) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {cpr 3641   U.cuni 3827    _I cid 4304    X. cxp 4687   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632    tX ctx 17255
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-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-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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-tx 17257
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