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Theorem ressms 20106
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 20034 . . 3  |-  ( K  e.  MetSp  ->  K  e.  *MetSp )
2 ressxms 20105 . . 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 2443 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 eqid 2443 . . . . . 6  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
64, 5msmet 20037 . . . . 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 19945 . . . 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 5128 . . . . 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 4977 . . . . . 6  |-  ( ( ( Base `  K
)  X.  ( Base `  K ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)
1211reseq2i 5112 . . . . 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 2463 . . . 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 2443 . . . . . . 7  |-  ( Ks  A )  =  ( Ks  A )
15 eqid 2443 . . . . . . 7  |-  ( dist `  K )  =  (
dist `  K )
1614, 15ressds 14357 . . . . . 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 3548 . . . . . . 7  |-  ( (
Base `  K )  i^i  A )  =  ( A  i^i  ( Base `  K ) )
1914, 4ressbas 14233 . . . . . . . 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 2487 . . . . . 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 5116 . . . 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 2487 . . 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 5700 . . 3  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  ( Met `  ( ( Base `  K )  i^i  A
) )  =  ( Met `  ( Base `  ( Ks  A ) ) ) )
269, 24, 253eltr3d 2523 . 2  |-  ( ( K  e.  MetSp  /\  A  e.  V )  ->  (
( dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( Met `  ( Base `  ( Ks  A ) ) ) )
27 eqid 2443 . . . 4  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2814, 27resstopn 18795 . . 3  |-  ( (
TopOpen `  K )t  A )  =  ( TopOpen `  ( Ks  A ) )
29 eqid 2443 . . 3  |-  ( Base `  ( Ks  A ) )  =  ( Base `  ( Ks  A ) )
30 eqid 2443 . . 3  |-  ( (
dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
3128, 29, 30isms 20029 . 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 1369    e. wcel 1756    i^i cin 3332    X. cxp 4843    |` cres 4847   ` cfv 5423  (class class class)co 6096   Basecbs 14179   ↾s cress 14180   distcds 14252   ↾t crest 14364   TopOpenctopn 14365   Metcme 17807   *MetSpcxme 19897   MetSpcmt 19898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-tset 14262  df-ds 14265  df-rest 14366  df-topn 14367  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-xms 19900  df-ms 19901
This theorem is referenced by:  subgngp  20226  cmsss  20866  cnpwstotbnd  28701
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