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Theorem ressxms 20080
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 20011 . . . . 5  |-  ( K  e.  *MetSp  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( *Met `  ( Base `  K
) ) )
43adantr 465 . . . 4  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( *Met `  ( Base `  K )
) )
5 xmetres 19919 . . . 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 5118 . . . . 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 4967 . . . . . 6  |-  ( ( ( Base `  K
)  X.  ( Base `  K ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)
98reseq2i 5102 . . . . 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 14344 . . . . . 6  |-  ( A  e.  V  ->  ( dist `  K )  =  ( dist `  ( Ks  A ) ) )
1413adantl 466 . . . . 5  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( dist `  K
)  =  ( dist `  ( Ks  A ) ) )
15 incom 3538 . . . . . . 7  |-  ( (
Base `  K )  i^i  A )  =  ( A  i^i  ( Base `  K ) )
1611, 1ressbas 14220 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  ( Base `  K
) )  =  (
Base `  ( Ks  A
) ) )
1716adantl 466 . . . . . . 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 4860 . . . . 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 5106 . . . 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 5690 . . 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 20006 . . . . . 6  |-  ( K  e.  *MetSp  ->  ( TopOpen
`  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
2625adantr 465 . . . . 5  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )
2726oveq1d 6101 . . . 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 3565 . . . . 5  |-  ( (
Base `  K )  i^i  A )  C_  ( Base `  K )
29 xpss12 4940 . . . . . . . . 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 672 . . . . . . . 8  |-  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)  C_  ( ( Base `  K )  X.  ( Base `  K
) )
31 resabs1 5134 . . . . . . . 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 20079 . . . . 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 662 . . . 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 20008 . . . . . . . . 9  |-  ( K  e.  *MetSp  ->  K  e.  TopSp )
401, 24tpsuni 18523 . . . . . . . . 9  |-  ( K  e.  TopSp  ->  ( Base `  K )  =  U. ( TopOpen `  K )
)
4139, 40syl 16 . . . . . . . 8  |-  ( K  e.  *MetSp  ->  ( Base `  K )  = 
U. ( TopOpen `  K
) )
4241adantr 465 . . . . . . 7  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( Base `  K
)  =  U. ( TopOpen
`  K ) )
4342ineq2d 3547 . . . . . 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 6102 . . . 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 18521 . . . . . 6  |-  ( K  e.  TopSp 
<->  ( TopOpen `  K )  e.  (TopOn `  ( Base `  K ) ) )
4739, 46sylib 196 . . . . 5  |-  ( K  e.  *MetSp  ->  ( TopOpen
`  K )  e.  (TopOn `  ( Base `  K ) ) )
48 eqid 2438 . . . . . 6  |-  U. ( TopOpen
`  K )  = 
U. ( TopOpen `  K
)
4948restin 18750 . . . . 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 471 . . . 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 5690 . . 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 18770 . . 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 20003 . 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 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 3322    C_ wss 3323   U.cuni 4086    X. cxp 4833    |` cres 4837   ` cfv 5413  (class class class)co 6086   Basecbs 14166   ↾s cress 14167   distcds 14239   ↾t crest 14351   TopOpenctopn 14352   *Metcxmt 17781   MetOpencmopn 17786  TopOnctopon 18479   TopSpctps 18481   *MetSpcxme 19872
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-tset 14249  df-ds 14252  df-rest 14353  df-topn 14354  df-topgen 14374  df-psmet 17789  df-xmet 17790  df-bl 17792  df-mopn 17793  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-xms 19875
This theorem is referenced by:  ressms  20081  qqhcn  26393  qqhucn  26394
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