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Theorem ressxms 21154
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 2457 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2457 . . . . . 6  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
31, 2xmsxmet 21085 . . . . 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 20993 . . . 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 5296 . . . . 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 5145 . . . . . 6  |-  ( ( ( Base `  K
)  X.  ( Base `  K ) )  i^i  ( A  X.  A
) )  =  ( ( ( Base `  K
)  i^i  A )  X.  ( ( Base `  K
)  i^i  A )
)
98reseq2i 5280 . . . . 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 2486 . . . 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 2457 . . . . . . 7  |-  ( Ks  A )  =  ( Ks  A )
12 eqid 2457 . . . . . . 7  |-  ( dist `  K )  =  (
dist `  K )
1311, 12ressds 14830 . . . . . 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 3687 . . . . . . 7  |-  ( (
Base `  K )  i^i  A )  =  ( A  i^i  ( Base `  K ) )
1611, 1ressbas 14701 . . . . . . . 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 2510 . . . . . 6  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( ( Base `  K )  i^i  A
)  =  ( Base `  ( Ks  A ) ) )
1918sqxpeqd 5034 . . . . 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 5284 . . . 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 2510 . . 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 5876 . . 3  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( *Met `  ( ( Base `  K
)  i^i  A )
)  =  ( *Met `  ( Base `  ( Ks  A ) ) ) )
236, 21, 223eltr3d 2559 . 2  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )  e.  ( *Met `  ( Base `  ( Ks  A ) ) ) )
24 eqid 2457 . . . . . . 7  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
2524, 1, 2xmstopn 21080 . . . . . 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 6311 . . . 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 3714 . . . . 5  |-  ( (
Base `  K )  i^i  A )  C_  ( Base `  K )
29 xpss12 5117 . . . . . . . . 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 5312 . . . . . . . 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 2489 . . . . . 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 2457 . . . . . 6  |-  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) )
35 eqid 2457 . . . . . 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 21153 . . . . 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 2498 . . 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 21082 . . . . . . . . 9  |-  ( K  e.  *MetSp  ->  K  e.  TopSp )
401, 24tpsuni 19566 . . . . . . . . 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 3696 . . . . . 6  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( A  i^i  ( Base `  K )
)  =  ( A  i^i  U. ( TopOpen `  K ) ) )
4415, 43syl5eq 2510 . . . . 5  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( ( Base `  K )  i^i  A
)  =  ( A  i^i  U. ( TopOpen `  K ) ) )
4544oveq2d 6312 . . . 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 19564 . . . . . 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 2457 . . . . . 6  |-  U. ( TopOpen
`  K )  = 
U. ( TopOpen `  K
)
4948restin 19794 . . . . 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 2501 . . 3  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  ( ( Base `  K )  i^i  A
) )  =  ( ( TopOpen `  K )t  A
) )
5221fveq2d 5876 . . 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 2506 . 2  |-  ( ( K  e.  *MetSp  /\  A  e.  V )  ->  ( ( TopOpen `  K )t  A )  =  (
MetOpen `  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) ) ) )
5411, 24resstopn 19814 . . 3  |-  ( (
TopOpen `  K )t  A )  =  ( TopOpen `  ( Ks  A ) )
55 eqid 2457 . . 3  |-  ( Base `  ( Ks  A ) )  =  ( Base `  ( Ks  A ) )
56 eqid 2457 . . 3  |-  ( (
dist `  ( Ks  A
) )  |`  (
( Base `  ( Ks  A
) )  X.  ( Base `  ( Ks  A ) ) ) )  =  ( ( dist `  ( Ks  A ) )  |`  ( ( Base `  ( Ks  A ) )  X.  ( Base `  ( Ks  A ) ) ) )
5754, 55, 56isxms2 21077 . 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 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   U.cuni 4251    X. cxp 5006    |` cres 5010   ` cfv 5594  (class class class)co 6296   Basecbs 14644   ↾s cress 14645   distcds 14721   ↾t crest 14838   TopOpenctopn 14839   *Metcxmt 18530   MetOpencmopn 18535  TopOnctopon 19522   TopSpctps 19524   *MetSpcxme 20946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-tset 14731  df-ds 14734  df-rest 14840  df-topn 14841  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-xms 20949
This theorem is referenced by:  ressms  21155  qqhcn  28133  qqhucn  28134
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