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Theorem kgencmp 17608
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 17603 . . . 4  |-  ( J  e.  Top  ->  (𝑘Gen `  J )  e.  Top )
21adantr 453 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  (𝑘Gen `  J )  e. 
Top )
3 kgenss 17606 . . . 4  |-  ( J  e.  Top  ->  J  C_  (𝑘Gen `  J ) )
43adantr 453 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  J  C_  (𝑘Gen `  J ) )
5 ssrest 17271 . . 3  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  J  C_  (𝑘Gen `  J ) )  -> 
( Jt  K )  C_  (
(𝑘Gen `  J )t  K ) )
62, 4, 5syl2anc 644 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K ) 
C_  ( (𝑘Gen `  J
)t 
K ) )
7 cmptop 17489 . . . . . 6  |-  ( ( Jt  K )  e.  Comp  -> 
( Jt  K )  e.  Top )
87adantl 454 . . . . 5  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( Jt  K )  e.  Top )
9 restrcl 17252 . . . . . 6  |-  ( ( Jt  K )  e.  Top  ->  ( J  e.  _V  /\  K  e.  _V )
)
109simprd 451 . . . . 5  |-  ( ( Jt  K )  e.  Top  ->  K  e.  _V )
118, 10syl 16 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  K  e.  _V )
12 restval 13685 . . . 4  |-  ( ( (𝑘Gen `  J )  e. 
Top  /\  K  e.  _V )  ->  ( (𝑘Gen `  J )t  K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
132, 11, 12syl2anc 644 . . 3  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  =  ran  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) )
14 simpr 449 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  ->  x  e.  (𝑘Gen `  J
) )
15 simplr 733 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( Jt  K )  e.  Comp )
16 kgeni 17600 . . . . . 6  |-  ( ( x  e.  (𝑘Gen `  J
)  /\  ( Jt  K
)  e.  Comp )  ->  ( x  i^i  K
)  e.  ( Jt  K ) )
1714, 15, 16syl2anc 644 . . . . 5  |-  ( ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  /\  x  e.  (𝑘Gen `  J ) )  -> 
( x  i^i  K
)  e.  ( Jt  K ) )
18 eqid 2442 . . . . 5  |-  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )  =  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) )
1917, 18fmptd 5922 . . . 4  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( x  e.  (𝑘Gen `  J )  |->  ( x  i^i  K ) ) : (𝑘Gen `  J
) --> ( Jt  K ) )
20 frn 5626 . . . 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 3368 . 2  |-  ( ( J  e.  Top  /\  ( Jt  K )  e.  Comp )  ->  ( (𝑘Gen `  J
)t 
K )  C_  ( Jt  K ) )
236, 22eqssd 3351 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 360    = wceq 1653    e. wcel 1727   _Vcvv 2962    i^i cin 3305    C_ wss 3306    e. cmpt 4291   ran crn 4908   -->wf 5479   ` cfv 5483  (class class class)co 6110   ↾t crest 13679   Topctop 16989   Compccmp 17480  𝑘Genckgen 17596
This theorem is referenced by:  kgencmp2  17609  kgenidm  17610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-recs 6662  df-rdg 6697  df-oadd 6757  df-er 6934  df-en 7139  df-fin 7142  df-fi 7445  df-rest 13681  df-topgen 13698  df-top 16994  df-bases 16996  df-topon 16997  df-cmp 17481  df-kgen 17597
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