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Theorem tmsxms 21158
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 21156 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  D  =  ( dist `  K
) )
31tmsbas 21155 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  X  =  ( Base `  K
) )
43fveq2d 5852 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  ( *Met `  X )  =  ( *Met `  ( Base `  K
) ) )
52, 4eleq12d 2536 . . . 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 3508 . . 3  |-  ( Base `  K )  C_  ( Base `  K )
8 xmetres2 21033 . . 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 660 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( *Met `  ( Base `  K
) ) )
10 xmetf 21001 . . . . . 6  |-  ( (
dist `  K )  e.  ( *Met `  ( Base `  K )
)  ->  ( dist `  K ) : ( ( Base `  K
)  X.  ( Base `  K ) ) --> RR* )
11 ffn 5713 . . . . . 6  |-  ( (
dist `  K ) : ( ( Base `  K )  X.  ( Base `  K ) ) -->
RR*  ->  ( dist `  K
)  Fn  ( (
Base `  K )  X.  ( Base `  K
) ) )
12 fnresdm 5672 . . . . . 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 2498 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  (
( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  D )
1514fveq2d 5852 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) )  =  ( MetOpen `  D
) )
16 eqid 2454 . . . 4  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
171, 16tmstopn 21157 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( MetOpen
`  D )  =  ( TopOpen `  K )
)
1815, 17eqtr2d 2496 . 2  |-  ( D  e.  ( *Met `  X )  ->  ( TopOpen
`  K )  =  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) ) )
19 eqid 2454 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
20 eqid 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2454 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
2219, 20, 21isxms2 21120 . 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 662 1  |-  ( D  e.  ( *Met `  X )  ->  K  e.  *MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    C_ wss 3461    X. cxp 4986    |` cres 4990    Fn wfn 5565   -->wf 5566   ` cfv 5570   RR*cxr 9616   Basecbs 14719   distcds 14796   TopOpenctopn 14914   *Metcxmt 18601   MetOpencmopn 18606   *MetSpcxme 20989  toMetSpctmt 20991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-tset 14806  df-ds 14809  df-rest 14915  df-topn 14916  df-topgen 14936  df-psmet 18609  df-xmet 18610  df-bl 18612  df-mopn 18613  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-xms 20992  df-tms 20994
This theorem is referenced by:  tmsms  21159  tmsxps  21208  tmsxpsmopn  21209  tmsxpsval  21210
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