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Theorem ressms 20792
Description: The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
ressms  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  MetSp )

Proof of Theorem ressms
StepHypRef Expression
1 msxms 20720 . . 3  |-  ( K  e.  MetSp  ->  K  e.  *MetSp )
2 ressxms 20791 . . 3  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  *MetSp )
31, 2sylan 471 . 2  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  *MetSp )
4 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2467 . . . . . 6  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
64, 5msmet 20723 . . . . 5  |-  ( K  e.  MetSp  ->  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( Met `  ( Base `  K ) ) )
76adantr 465 . . . 4  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( Met `  ( Base `  K ) ) )
8 metres 20631 . . . 4  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( Met `  ( Base `  K
) )  ->  (
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  e.  ( Met `  (
( Base `  K )  i^i  A ) ) )
97, 8syl 16 . . 3  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  e.  ( Met `  (
( Base `  K )  i^i  A ) ) )
10 resres 5286 . . . . 5  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  =  ( ( dist `  K
)  |`  ( ( (
Base `  K )  X.  ( Base `  K
) )  i^i  ( A  X.  A ) ) )
11 inxp 5135 . . . . . 6  |-  ( ( ( Base `  K
)  X.  ( Base `  K ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)
1211reseq2i 5270 . . . . 5  |-  ( (
dist `  K )  |`  ( ( ( Base `  K )  X.  ( Base `  K ) )  i^i  ( A  X.  A ) ) )  =  ( ( dist `  K )  |`  (
( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
) )
1310, 12eqtri 2496 . . . 4  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  =  ( ( dist `  K
)  |`  ( ( (
Base `  K )  i^i  A )  X.  (
( Base `  K )  i^i  A ) ) )
14 eqid 2467 . . . . . . 7  |-  ( Ks  A )  =  ( Ks  A )
15 eqid 2467 . . . . . . 7  |-  ( dist `  K )  =  (
dist `  K )
1614, 15ressds 14669 . . . . . 6  |-  ( A  e.  V  ->  ( dist `  K )  =  ( dist `  ( Ks  A ) ) )
1716adantl 466 . . . . 5  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( dist `  K )  =  ( dist `  ( Ks  A ) ) )
18 incom 3691 . . . . . . 7  |-  ( (
Base `  K )  i^i  A )  =  ( A  i^i  ( Base `  K ) )
1914, 4ressbas 14545 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  A
) ) )
2019adantl 466 . . . . . . 7  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( A  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  A
) ) )
2118, 20syl5eq 2520 . . . . . 6  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( Base `  K )  i^i  A )  =  (
Base `  ( Ks  A
) ) )
2221, 21xpeq12d 5024 . . . . 5  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)  =  ( (
Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )
2317, 22reseq12d 5274 . . . 4  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( dist `  K )  |`  ( ( ( Base `  K )  i^i  A
)  X.  ( (
Base `  K )  i^i  A ) ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) ) )
2413, 23syl5eq 2520 . . 3  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) ) )
2521fveq2d 5870 . . 3  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( Met `  ( ( Base `  K )  i^i  A
) )  =  ( Met `  ( Base `  ( Ks  A ) ) ) )
269, 24, 253eltr3d 2569 . 2  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( Met `  ( Base `  ( Ks  A ) ) ) )
27 eqid 2467 . . . 4  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2814, 27resstopn 19481 . . 3  |-  ( (
TopOpen `  K )t  A )  =  ( TopOpen `  ( Ks  A ) )
29 eqid 2467 . . 3  |-  ( Base `  ( Ks  A ) )  =  ( Base `  ( Ks  A ) )
30 eqid 2467 . . 3  |-  ( (
dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
3128, 29, 30isms 20715 . 2  |-  ( ( Ks  A )  e.  MetSp  <->  (
( Ks  A )  e.  *MetSp  /\  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( Met `  ( Base `  ( Ks  A ) ) ) ) )
323, 26, 31sylanbrc 664 1  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3475    X. cxp 4997    |` cres 5001   ` cfv 5588  (class class class)co 6284   Basecbs 14490   ↾s cress 14491   distcds 14564   ↾t crest 14676   TopOpenctopn 14677   Metcme 18203   *MetSpcxme 20583   MetSpcmt 20584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-tset 14574  df-ds 14577  df-rest 14678  df-topn 14679  df-topgen 14699  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-xms 20586  df-ms 20587
This theorem is referenced by:  subgngp  20912  cmsss  21552  cnpwstotbnd  29924
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