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Theorem ressxms 18167
Description: The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
ressxms  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  * MetSp )

Proof of Theorem ressxms
StepHypRef Expression
1 resres 5047 . . . . 5  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  =  ( ( dist `  K
)  |`  ( ( (
Base `  K )  X.  ( Base `  K
) )  i^i  ( A  X.  A ) ) )
2 inxp 4897 . . . . . 6  |-  ( ( ( Base `  K
)  X.  ( Base `  K ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)
32reseq2i 5031 . . . . 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 )
) )
41, 3eqtri 2378 . . . 4  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  =  ( ( dist `  K
)  |`  ( ( (
Base `  K )  i^i  A )  X.  (
( Base `  K )  i^i  A ) ) )
5 eqid 2358 . . . . . . 7  |-  ( Ks  A )  =  ( Ks  A )
6 eqid 2358 . . . . . . 7  |-  ( dist `  K )  =  (
dist `  K )
75, 6ressds 13411 . . . . . 6  |-  ( A  e.  V  ->  ( dist `  K )  =  ( dist `  ( Ks  A ) ) )
87adantl 452 . . . . 5  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( dist `  K
)  =  ( dist `  ( Ks  A ) ) )
9 incom 3437 . . . . . . 7  |-  ( (
Base `  K )  i^i  A )  =  ( A  i^i  ( Base `  K ) )
10 eqid 2358 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
115, 10ressbas 13289 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  A
) ) )
1211adantl 452 . . . . . . 7  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( A  i^i  ( Base `  K )
)  =  ( Base `  ( Ks  A ) ) )
139, 12syl5eq 2402 . . . . . 6  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( Base `  K )  i^i  A
)  =  ( Base `  ( Ks  A ) ) )
1413, 13xpeq12d 4793 . . . . 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 ) ) ) )
158, 14reseq12d 5035 . . . 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 ) ) ) ) )
164, 15syl5eq 2402 . . 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 ) ) ) ) )
17 eqid 2358 . . . . . . 7  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1810, 17xmsxmet 18098 . . . . . 6  |-  ( K  e.  * MetSp  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( * Met `  ( Base `  K
) ) )
1918adantr 451 . . . . 5  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( * Met `  ( Base `  K ) ) )
20 xmetres 18024 . . . . 5  |-  ( ( ( 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 ) ) )
2119, 20syl 15 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  |`  ( A  X.  A
) )  e.  ( * Met `  (
( Base `  K )  i^i  A ) ) )
2213fveq2d 5609 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( * Met `  ( ( Base `  K
)  i^i  A )
)  =  ( * Met `  ( Base `  ( Ks  A ) ) ) )
2321, 22eleqtrd 2434 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  |`  ( A  X.  A
) )  e.  ( * Met `  ( Base `  ( Ks  A ) ) ) )
2416, 23eqeltrrd 2433 . 2  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( * Met `  ( Base `  ( Ks  A ) ) ) )
25 eqid 2358 . . . . . . 7  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2625, 10, 17xmstopn 18093 . . . . . 6  |-  ( K  e.  * MetSp  ->  ( TopOpen
`  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
2726adantr 451 . . . . 5  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )
2827oveq1d 5957 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  ( ( Base `  K )  i^i  A
) )  =  ( ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )t  ( ( Base `  K
)  i^i  A )
) )
29 inss1 3465 . . . . 5  |-  ( (
Base `  K )  i^i  A )  C_  ( Base `  K )
30 xpss12 4871 . . . . . . . . 9  |-  ( ( ( ( Base `  K
)  i^i  A )  C_  ( Base `  K
)  /\  ( ( Base `  K )  i^i 
A )  C_  ( Base `  K ) )  ->  ( ( (
Base `  K )  i^i  A )  X.  (
( Base `  K )  i^i  A ) )  C_  ( ( Base `  K
)  X.  ( Base `  K ) ) )
3129, 29, 30mp2an 653 . . . . . . . 8  |-  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)  C_  ( ( Base `  K )  X.  ( Base `  K
) )
32 resabs1 5063 . . . . . . . 8  |-  ( ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)  C_  ( ( Base `  K )  X.  ( Base `  K
) )  ->  (
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( (
( Base `  K )  i^i  A )  X.  (
( Base `  K )  i^i  A ) ) )  =  ( ( dist `  K )  |`  (
( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
) ) )
3331, 32ax-mp 8 . . . . . . 7  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( (
( Base `  K )  i^i  A )  X.  (
( Base `  K )  i^i  A ) ) )  =  ( ( dist `  K )  |`  (
( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
) )
344, 33eqtr4i 2381 . . . . . 6  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  =  ( ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  |`  ( ( ( Base `  K )  i^i  A
)  X.  ( (
Base `  K )  i^i  A ) ) )
35 eqid 2358 . . . . . 6  |-  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) )
36 eqid 2358 . . . . . 6  |-  ( MetOpen `  ( ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  |`  ( A  X.  A
) ) )  =  ( MetOpen `  ( (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  |`  ( A  X.  A
) ) )
3734, 35, 36metrest 18166 . . . . 5  |-  ( ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( * Met `  ( Base `  K ) )  /\  ( ( Base `  K
)  i^i  A )  C_  ( Base `  K
) )  ->  (
( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )t  ( ( Base `  K
)  i^i  A )
)  =  ( MetOpen `  ( ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  |`  ( A  X.  A
) ) ) )
3819, 29, 37sylancl 643 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )t  ( ( Base `  K )  i^i  A
) )  =  (
MetOpen `  ( ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  |`  ( A  X.  A
) ) ) )
3928, 38eqtrd 2390 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  ( ( Base `  K )  i^i  A
) )  =  (
MetOpen `  ( ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  |`  ( A  X.  A
) ) ) )
40 xmstps 18095 . . . . . . . . 9  |-  ( K  e.  * MetSp  ->  K  e.  TopSp )
4110, 25tpsuni 16776 . . . . . . . . 9  |-  ( K  e.  TopSp  ->  ( Base `  K )  =  U. ( TopOpen `  K )
)
4240, 41syl 15 . . . . . . . 8  |-  ( K  e.  * MetSp  ->  ( Base `  K )  = 
U. ( TopOpen `  K
) )
4342adantr 451 . . . . . . 7  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( Base `  K
)  =  U. ( TopOpen
`  K ) )
4443ineq2d 3446 . . . . . 6  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( A  i^i  ( Base `  K )
)  =  ( A  i^i  U. ( TopOpen `  K ) ) )
459, 44syl5eq 2402 . . . . 5  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( Base `  K )  i^i  A
)  =  ( A  i^i  U. ( TopOpen `  K ) ) )
4645oveq2d 5958 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  ( ( Base `  K )  i^i  A
) )  =  ( ( TopOpen `  K )t  ( A  i^i  U. ( TopOpen `  K ) ) ) )
4710, 25istps 16774 . . . . . 6  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) ) )
4840, 47sylib 188 . . . . 5  |-  ( K  e.  * MetSp  ->  ( TopOpen
`  K )  e.  (TopOn `  ( Base `  K ) ) )
49 eqid 2358 . . . . . 6  |-  U. ( TopOpen
`  K )  = 
U. ( TopOpen `  K
)
5049restin 16997 . . . . 5  |-  ( ( ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) )  /\  A  e.  V )  ->  ( ( TopOpen `  K
)t 
A )  =  ( ( TopOpen `  K )t  ( A  i^i  U. ( TopOpen `  K ) ) ) )
5148, 50sylan 457 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  A )  =  ( ( TopOpen `  K )t  ( A  i^i  U. ( TopOpen `  K ) ) ) )
5246, 51eqtr4d 2393 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  ( ( Base `  K )  i^i  A
) )  =  ( ( TopOpen `  K )t  A
) )
5316fveq2d 5609 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( MetOpen `  (
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) ) )  =  ( MetOpen `  (
( dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) ) ) )
5439, 52, 533eqtr3d 2398 . 2  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  A )  =  (
MetOpen `  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) ) ) )
555, 25resstopn 17016 . . 3  |-  ( (
TopOpen `  K )t  A )  =  ( TopOpen `  ( Ks  A ) )
56 eqid 2358 . . 3  |-  ( Base `  ( Ks  A ) )  =  ( Base `  ( Ks  A ) )
57 eqid 2358 . . 3  |-  ( (
dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
5855, 56, 57isxms2 18090 . 2  |-  ( ( Ks  A )  e.  * MetSp  <-> 
( ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( * Met `  ( Base `  ( Ks  A ) ) )  /\  ( ( TopOpen `  K )t  A )  =  (
MetOpen `  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) ) ) ) )
5924, 54, 58sylanbrc 645 1  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  * MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   U.cuni 3906    X. cxp 4766    |` cres 4770   ` cfv 5334  (class class class)co 5942   Basecbs 13239   ↾s cress 13240   distcds 13308   ↾t crest 13418   TopOpenctopn 13419   * Metcxmt 16462   MetOpencmopn 16467  TopOnctopon 16732   TopSpctps 16734   *
MetSpcxme 17978
This theorem is referenced by:  ressms  18168  qqhcn  23648  dya2icoseg2  23892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-tset 13318  df-ds 13321  df-rest 13420  df-topn 13421  df-topgen 13437  df-xmet 16469  df-bl 16471  df-mopn 16472  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-xms 17981
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