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Theorem ressxms 18560
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 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2438 . . . . . 6  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
31, 2xmsxmet 18491 . . . . 5  |-  ( K  e.  * MetSp  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( * Met `  ( Base `  K
) ) )
43adantr 453 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( * Met `  ( Base `  K ) ) )
5 xmetres 18399 . . . 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 ) ) )
64, 5syl 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 ) ) )
7 resres 5162 . . . . 5  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  =  ( ( dist `  K
)  |`  ( ( (
Base `  K )  X.  ( Base `  K
) )  i^i  ( A  X.  A ) ) )
8 inxp 5010 . . . . . 6  |-  ( ( ( Base `  K
)  X.  ( Base `  K ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)
98reseq2i 5146 . . . . 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 )
) )
107, 9eqtri 2458 . . . 4  |-  ( ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  |`  ( A  X.  A ) )  =  ( ( dist `  K
)  |`  ( ( (
Base `  K )  i^i  A )  X.  (
( Base `  K )  i^i  A ) ) )
11 eqid 2438 . . . . . . 7  |-  ( Ks  A )  =  ( Ks  A )
12 eqid 2438 . . . . . . 7  |-  ( dist `  K )  =  (
dist `  K )
1311, 12ressds 13646 . . . . . 6  |-  ( A  e.  V  ->  ( dist `  K )  =  ( dist `  ( Ks  A ) ) )
1413adantl 454 . . . . 5  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( dist `  K
)  =  ( dist `  ( Ks  A ) ) )
15 incom 3535 . . . . . . 7  |-  ( (
Base `  K )  i^i  A )  =  ( A  i^i  ( Base `  K ) )
1611, 1ressbas 13524 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  A
) ) )
1716adantl 454 . . . . . . 7  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( A  i^i  ( Base `  K )
)  =  ( Base `  ( Ks  A ) ) )
1815, 17syl5eq 2482 . . . . . 6  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( Base `  K )  i^i  A
)  =  ( Base `  ( Ks  A ) ) )
1918, 18xpeq12d 4906 . . . . 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 ) ) ) )
2014, 19reseq12d 5150 . . . 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 ) ) ) ) )
2110, 20syl5eq 2482 . . 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 ) ) ) ) )
2218fveq2d 5735 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( * Met `  ( ( Base `  K
)  i^i  A )
)  =  ( * Met `  ( Base `  ( Ks  A ) ) ) )
236, 21, 223eltr3d 2518 . 2  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( * Met `  ( Base `  ( Ks  A ) ) ) )
24 eqid 2438 . . . . . . 7  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2524, 1, 2xmstopn 18486 . . . . . 6  |-  ( K  e.  * MetSp  ->  ( TopOpen
`  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
2625adantr 453 . . . . 5  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )
2726oveq1d 6099 . . . 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 )
) )
28 inss1 3563 . . . . 5  |-  ( (
Base `  K )  i^i  A )  C_  ( Base `  K )
29 xpss12 4984 . . . . . . . . 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 ) ) )
3028, 28, 29mp2an 655 . . . . . . . 8  |-  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)  C_  ( ( Base `  K )  X.  ( Base `  K
) )
31 resabs1 5178 . . . . . . . 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 )
) ) )
3230, 31ax-mp 5 . . . . . . 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 )
) )
3310, 32eqtr4i 2461 . . . . . 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 ) ) )
34 eqid 2438 . . . . . 6  |-  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) )
35 eqid 2438 . . . . . 6  |-  ( MetOpen `  ( ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  |`  ( A  X.  A
) ) )  =  ( MetOpen `  ( (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  |`  ( A  X.  A
) ) )
3633, 34, 35metrest 18559 . . . . 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
) ) ) )
374, 28, 36sylancl 645 . . . 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
) ) ) )
3827, 37eqtrd 2470 . . 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
) ) ) )
39 xmstps 18488 . . . . . . . . 9  |-  ( K  e.  * MetSp  ->  K  e.  TopSp )
401, 24tpsuni 17008 . . . . . . . . 9  |-  ( K  e.  TopSp  ->  ( Base `  K )  =  U. ( TopOpen `  K )
)
4139, 40syl 16 . . . . . . . 8  |-  ( K  e.  * MetSp  ->  ( Base `  K )  = 
U. ( TopOpen `  K
) )
4241adantr 453 . . . . . . 7  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( Base `  K
)  =  U. ( TopOpen
`  K ) )
4342ineq2d 3544 . . . . . 6  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( A  i^i  ( Base `  K )
)  =  ( A  i^i  U. ( TopOpen `  K ) ) )
4415, 43syl5eq 2482 . . . . 5  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( Base `  K )  i^i  A
)  =  ( A  i^i  U. ( TopOpen `  K ) ) )
4544oveq2d 6100 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  ( ( Base `  K )  i^i  A
) )  =  ( ( TopOpen `  K )t  ( A  i^i  U. ( TopOpen `  K ) ) ) )
461, 24istps 17006 . . . . . 6  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) ) )
4739, 46sylib 190 . . . . 5  |-  ( K  e.  * MetSp  ->  ( TopOpen
`  K )  e.  (TopOn `  ( Base `  K ) ) )
48 eqid 2438 . . . . . 6  |-  U. ( TopOpen
`  K )  = 
U. ( TopOpen `  K
)
4948restin 17235 . . . . 5  |-  ( ( ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) )  /\  A  e.  V )  ->  ( ( TopOpen `  K
)t 
A )  =  ( ( TopOpen `  K )t  ( A  i^i  U. ( TopOpen `  K ) ) ) )
5047, 49sylan 459 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  A )  =  ( ( TopOpen `  K )t  ( A  i^i  U. ( TopOpen `  K ) ) ) )
5145, 50eqtr4d 2473 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  ( ( Base `  K )  i^i  A
) )  =  ( ( TopOpen `  K )t  A
) )
5221fveq2d 5735 . . 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 ) ) ) ) ) )
5338, 51, 523eqtr3d 2478 . 2  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  A )  =  (
MetOpen `  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) ) ) )
5411, 24resstopn 17255 . . 3  |-  ( (
TopOpen `  K )t  A )  =  ( TopOpen `  ( Ks  A ) )
55 eqid 2438 . . 3  |-  ( Base `  ( Ks  A ) )  =  ( Base `  ( Ks  A ) )
56 eqid 2438 . . 3  |-  ( (
dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
5754, 55, 56isxms2 18483 . 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 ) ) ) ) ) ) )
5823, 53, 57sylanbrc 647 1  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( Ks  A )  e.  * MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   U.cuni 4017    X. cxp 4879    |` cres 4883   ` cfv 5457  (class class class)co 6084   Basecbs 13474   ↾s cress 13475   distcds 13543   ↾t crest 13653   TopOpenctopn 13654   * Metcxmt 16691   MetOpencmopn 16696  TopOnctopon 16964   TopSpctps 16966   *
MetSpcxme 18352
This theorem is referenced by:  ressms  18561  qqhcn  24380  qqhucn  24381  dya2icoseg2  24633
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 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-tset 13553  df-ds 13556  df-rest 13655  df-topn 13656  df-topgen 13672  df-psmet 16699  df-xmet 16700  df-bl 16702  df-mopn 16703  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-xms 18355
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