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Theorem ressms 18517
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 18445 . . 3  |-  ( K  e.  MetSp  ->  K  e.  *
MetSp )
2 ressxms 18516 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  * MetSp )
31, 2sylan 458 . 2  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  * MetSp )
4 eqid 2412 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2412 . . . . . 6  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
64, 5msmet 18448 . . . . 5  |-  ( K  e.  MetSp  ->  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( Met `  ( Base `  K ) ) )
76adantr 452 . . . 4  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( Met `  ( Base `  K ) ) )
8 metres 18356 . . . 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 5126 . . . . 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 4974 . . . . . 6  |-  ( ( ( Base `  K
)  X.  ( Base `  K ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)
1211reseq2i 5110 . . . . 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 2432 . . . 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 2412 . . . . . . 7  |-  ( Ks  A )  =  ( Ks  A )
15 eqid 2412 . . . . . . 7  |-  ( dist `  K )  =  (
dist `  K )
1614, 15ressds 13604 . . . . . 6  |-  ( A  e.  V  ->  ( dist `  K )  =  ( dist `  ( Ks  A ) ) )
1716adantl 453 . . . . 5  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( dist `  K )  =  ( dist `  ( Ks  A ) ) )
18 incom 3501 . . . . . . 7  |-  ( (
Base `  K )  i^i  A )  =  ( A  i^i  ( Base `  K ) )
1914, 4ressbas 13482 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  A
) ) )
2019adantl 453 . . . . . . 7  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( A  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  A
) ) )
2118, 20syl5eq 2456 . . . . . 6  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( Base `  K )  i^i  A )  =  (
Base `  ( Ks  A
) ) )
2221, 21xpeq12d 4870 . . . . 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 5114 . . . 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 2456 . . 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 5699 . . 3  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( Met `  ( ( Base `  K )  i^i  A
) )  =  ( Met `  ( Base `  ( Ks  A ) ) ) )
269, 24, 253eltr3d 2492 . 2  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( Met `  ( Base `  ( Ks  A ) ) ) )
27 eqid 2412 . . . 4  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2814, 27resstopn 17212 . . 3  |-  ( (
TopOpen `  K )t  A )  =  ( TopOpen `  ( Ks  A ) )
29 eqid 2412 . . 3  |-  ( Base `  ( Ks  A ) )  =  ( Base `  ( Ks  A ) )
30 eqid 2412 . . 3  |-  ( (
dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
3128, 29, 30isms 18440 . 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 646 1  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3287    X. cxp 4843    |` cres 4847   ` cfv 5421  (class class class)co 6048   Basecbs 13432   ↾s cress 13433   distcds 13501   ↾t crest 13611   TopOpenctopn 13612   Metcme 16650   *
MetSpcxme 18308   MetSpcmt 18309
This theorem is referenced by:  subgngp  18637  cmsss  19264  cnpwstotbnd  26404
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  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  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-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-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-tset 13511  df-ds 13514  df-rest 13613  df-topn 13614  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-xms 18311  df-ms 18312
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