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

Theorem kgencmp 17538
Description: The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgencmp  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  =  ( (𝑘Gen `  J
)t 
K ) )

Proof of Theorem kgencmp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 kgenftop 17533 . . . 4  |-  ( J  e.  Top  ->  (𝑘Gen `  J )  e.  Top )
21adantr 452 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  (𝑘Gen `  J )  e. 
Top )
3 kgenss 17536 . . . 4  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
43adantr 452 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  J  C_  (𝑘Gen `  J ) )
5 ssrest 17202 . . 3  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  J  C_  (𝑘Gen `  J ) )  -> 
( Jt  K )  C_  (
(𝑘Gen `  J )t  K ) )
62, 4, 5syl2anc 643 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K ) 
C_  ( (𝑘Gen `  J
)t 
K ) )
7 cmptop 17420 . . . . . 6  |-  ( ( Jt  K )  e.  Comp  -> 
( Jt  K )  e.  Top )
87adantl 453 . . . . 5  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  e.  Top )
9 restrcl 17183 . . . . . 6  |-  ( ( Jt  K )  e.  Top  ->  ( J  e.  _V  /\  K  e.  _V )
)
109simprd 450 . . . . 5  |-  ( ( Jt  K )  e.  Top  ->  K  e.  _V )
118, 10syl 16 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  K  e.  _V )
12 restval 13617 . . . 4  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  K  e.  _V )  ->  ( (𝑘Gen `  J )t  K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
132, 11, 12syl2anc 643 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
14 simpr 448 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  ->  x  e.  (𝑘Gen `  J
) )
15 simplr 732 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( Jt  K )  e.  Comp )
16 kgeni 17530 . . . . . 6  |-  ( ( x  e.  (𝑘Gen `  J
)  /\  ( Jt  K
)  e.  Comp )  ->  ( x  i^i  K
)  e.  ( Jt  K ) )
1714, 15, 16syl2anc 643 . . . . 5  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( x  i^i  K
)  e.  ( Jt  K ) )
18 eqid 2412 . . . . 5  |-  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )  =  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )
1917, 18fmptd 5860 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) : (𝑘Gen `  J
) --> ( Jt  K ) )
20 frn 5564 . . . 4  |-  ( ( x  e.  (𝑘Gen `  J
)  |->  ( x  i^i 
K ) ) : (𝑘Gen `  J ) --> ( Jt  K )  ->  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) 
C_  ( Jt  K ) )
2119, 20syl 16 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )  C_  ( Jt  K
) )
2213, 21eqsstrd 3350 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  C_  ( Jt  K ) )
236, 22eqssd 3333 1  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  =  ( (𝑘Gen `  J
)t 
K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2924    i^i cin 3287    C_ wss 3288    e. cmpt 4234   ran crn 4846   -->wf 5417   ` cfv 5421  (class class class)co 6048   ↾t crest 13611   Topctop 16921   Compccmp 17411  𝑘Genckgen 17526
This theorem is referenced by:  kgencmp2  17539  kgenidm  17540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-oadd 6695  df-er 6872  df-en 7077  df-fin 7080  df-fi 7382  df-rest 13613  df-topgen 13630  df-top 16926  df-bases 16928  df-topon 16929  df-cmp 17412  df-kgen 17527
  Copyright terms: Public domain W3C validator