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Theorem tmsxms 20855
Description: The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypothesis
Ref Expression
tmsbas.k  |-  K  =  (toMetSp `  D )
Assertion
Ref Expression
tmsxms  |-  ( D  e.  ( *Met `  X )  ->  K  e.  *MetSp )

Proof of Theorem tmsxms
StepHypRef Expression
1 tmsbas.k . . . . . 6  |-  K  =  (toMetSp `  D )
21tmsds 20853 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  D  =  ( dist `  K
) )
31tmsbas 20852 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  X  =  ( Base `  K
) )
43fveq2d 5856 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  ( *Met `  X )  =  ( *Met `  ( Base `  K
) ) )
52, 4eleq12d 2523 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( D  e.  ( *Met `  X )  <->  ( dist `  K )  e.  ( *Met `  ( Base `  K ) ) ) )
65ibi 241 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( dist `  K )  e.  ( *Met `  ( Base `  K )
) )
7 ssid 3505 . . 3  |-  ( Base `  K )  C_  ( Base `  K )
8 xmetres2 20730 . . 3  |-  ( ( ( dist `  K
)  e.  ( *Met `  ( Base `  K ) )  /\  ( Base `  K )  C_  ( Base `  K
) )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( *Met `  ( Base `  K
) ) )
96, 7, 8sylancl 662 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( *Met `  ( Base `  K
) ) )
10 xmetf 20698 . . . . . 6  |-  ( (
dist `  K )  e.  ( *Met `  ( Base `  K )
)  ->  ( dist `  K ) : ( ( Base `  K
)  X.  ( Base `  K ) ) --> RR* )
11 ffn 5717 . . . . . 6  |-  ( (
dist `  K ) : ( ( Base `  K )  X.  ( Base `  K ) ) -->
RR*  ->  ( dist `  K
)  Fn  ( (
Base `  K )  X.  ( Base `  K
) ) )
12 fnresdm 5676 . . . . . 6  |-  ( (
dist `  K )  Fn  ( ( Base `  K
)  X.  ( Base `  K ) )  -> 
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( dist `  K ) )
136, 10, 11, 124syl 21 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( dist `  K
) )
1413, 2eqtr4d 2485 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  D )
1514fveq2d 5856 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) )  =  ( MetOpen `  D
) )
16 eqid 2441 . . . 4  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
171, 16tmstopn 20854 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  D )  =  ( TopOpen `  K )
)
1815, 17eqtr2d 2483 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( TopOpen
`  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
19 eqid 2441 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
20 eqid 2441 . . 3  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2441 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
2219, 20, 21isxms2 20817 . 2  |-  ( K  e.  *MetSp  <->  ( (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( *Met `  ( Base `  K
) )  /\  ( TopOpen
`  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) ) )
239, 18, 22sylanbrc 664 1  |-  ( D  e.  ( *Met `  X )  ->  K  e.  *MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802    C_ wss 3458    X. cxp 4983    |` cres 4987    Fn wfn 5569   -->wf 5570   ` cfv 5574   RR*cxr 9625   Basecbs 14504   distcds 14578   TopOpenctopn 14691   *Metcxmt 18271   MetOpencmopn 18276   *MetSpcxme 20686  toMetSpctmt 20688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-fz 11677  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-tset 14588  df-ds 14591  df-rest 14692  df-topn 14693  df-topgen 14713  df-psmet 18279  df-xmet 18280  df-bl 18282  df-mopn 18283  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-xms 20689  df-tms 20691
This theorem is referenced by:  tmsms  20856  tmsxps  20905  tmsxpsmopn  20906  tmsxpsval  20907
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