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Theorem ressxms 18071
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 4968 . . . . 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 4818 . . . . . 6  |-  ( ( ( Base `  K
)  X.  ( Base `  K ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)
32reseq2i 4952 . . . . 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 2303 . . . 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 2283 . . . . . . 7  |-  ( Ks  A )  =  ( Ks  A )
6 eqid 2283 . . . . . . 7  |-  ( dist `  K )  =  (
dist `  K )
75, 6ressds 13318 . . . . . 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 3361 . . . . . . 7  |-  ( (
Base `  K )  i^i  A )  =  ( A  i^i  ( Base `  K ) )
10 eqid 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
115, 10ressbas 13198 . . . . . . . 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 2327 . . . . . 6  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( Base `  K )  i^i  A
)  =  ( Base `  ( Ks  A ) ) )
1413, 13xpeq12d 4714 . . . . 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 4956 . . . 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 2327 . . 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 2283 . . . . . . 7  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1810, 17xmsxmet 18002 . . . . . 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 17928 . . . . 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 5529 . . . 4  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( * Met `  ( ( Base `  K
)  i^i  A )
)  =  ( * Met `  ( Base `  ( Ks  A ) ) ) )
2321, 22eleqtrd 2359 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  |`  ( A  X.  A
) )  e.  ( * Met `  ( Base `  ( Ks  A ) ) ) )
2416, 23eqeltrrd 2358 . 2  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( * Met `  ( Base `  ( Ks  A ) ) ) )
25 eqid 2283 . . . . . . 7  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2625, 10, 17xmstopn 17997 . . . . . 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 5873 . . . 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 3389 . . . . 5  |-  ( (
Base `  K )  i^i  A )  C_  ( Base `  K )
30 xpss12 4792 . . . . . . . . 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 4984 . . . . . . . 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 2306 . . . . . 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 2283 . . . . . 6  |-  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) )
36 eqid 2283 . . . . . 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 18070 . . . . 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 2315 . . 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 17999 . . . . . . . . 9  |-  ( K  e.  * MetSp  ->  K  e.  TopSp )
4110, 25tpsuni 16676 . . . . . . . . 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 3370 . . . . . 6  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( A  i^i  ( Base `  K )
)  =  ( A  i^i  U. ( TopOpen `  K ) ) )
459, 44syl5eq 2327 . . . . 5  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( Base `  K )  i^i  A
)  =  ( A  i^i  U. ( TopOpen `  K ) ) )
4645oveq2d 5874 . . . 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 16674 . . . . . 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 2283 . . . . . 6  |-  U. ( TopOpen
`  K )  = 
U. ( TopOpen `  K
)
5049restin 16897 . . . . 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 2318 . . 3  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  ( ( Base `  K )  i^i  A
) )  =  ( ( TopOpen `  K )t  A
) )
5316fveq2d 5529 . . 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 2323 . 2  |-  ( ( K  e.  * MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  A )  =  (
MetOpen `  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) ) ) )
555, 25resstopn 16916 . . 3  |-  ( (
TopOpen `  K )t  A )  =  ( TopOpen `  ( Ks  A ) )
56 eqid 2283 . . 3  |-  ( Base `  ( Ks  A ) )  =  ( Base `  ( Ks  A ) )
57 eqid 2283 . . 3  |-  ( (
dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
5855, 56, 57isxms2 17994 . 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 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   U.cuni 3827    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   distcds 13217   ↾t crest 13325   TopOpenctopn 13326   * Metcxmt 16369   MetOpencmopn 16372  TopOnctopon 16632   TopSpctps 16634   *
MetSpcxme 17882
This theorem is referenced by:  ressms  18072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-tset 13227  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885
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