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Theorem tmsxms 20186
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 20184 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  D  =  ( dist `  K
) )
31tmsbas 20183 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  X  =  ( Base `  K
) )
43fveq2d 5796 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  ( *Met `  X )  =  ( *Met `  ( Base `  K
) ) )
52, 4eleq12d 2533 . . . 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 3476 . . 3  |-  ( Base `  K )  C_  ( Base `  K )
8 xmetres2 20061 . . 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 20029 . . . . . 6  |-  ( (
dist `  K )  e.  ( *Met `  ( Base `  K )
)  ->  ( dist `  K ) : ( ( Base `  K
)  X.  ( Base `  K ) ) --> RR* )
11 ffn 5660 . . . . . 6  |-  ( (
dist `  K ) : ( ( Base `  K )  X.  ( Base `  K ) ) -->
RR*  ->  ( dist `  K
)  Fn  ( (
Base `  K )  X.  ( Base `  K
) ) )
12 fnresdm 5621 . . . . . 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 2495 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  D )
1514fveq2d 5796 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) )  =  ( MetOpen `  D
) )
16 eqid 2451 . . . 4  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
171, 16tmstopn 20185 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  D )  =  ( TopOpen `  K )
)
1815, 17eqtr2d 2493 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( TopOpen
`  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
19 eqid 2451 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
20 eqid 2451 . . 3  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2451 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
2219, 20, 21isxms2 20148 . 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 1370    e. wcel 1758    C_ wss 3429    X. cxp 4939    |` cres 4943    Fn wfn 5514   -->wf 5515   ` cfv 5519   RR*cxr 9521   Basecbs 14285   distcds 14358   TopOpenctopn 14471   *Metcxmt 17919   MetOpencmopn 17924   *MetSpcxme 20017  toMetSpctmt 20019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-tset 14368  df-ds 14371  df-rest 14472  df-topn 14473  df-topgen 14493  df-psmet 17927  df-xmet 17928  df-bl 17930  df-mopn 17931  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-xms 20020  df-tms 20022
This theorem is referenced by:  tmsms  20187  tmsxps  20236  tmsxpsmopn  20237  tmsxpsval  20238
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