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Theorem prdsxms 21199
Description: The indexed product structure is an extended metric space when the index set is finite. (Although the extended metric is still valid when the index set is infinite, it no longer agrees with the product topology, which is not metrizable in any case.) (Contributed by Mario Carneiro, 28-Aug-2015.)
Hypothesis
Ref Expression
prdsxms.y  |-  Y  =  ( S X_s R )
Assertion
Ref Expression
prdsxms  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  Y  e.  *MetSp )

Proof of Theorem prdsxms
Dummy variables  g 
k  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsxms.y . . . 4  |-  Y  =  ( S X_s R )
2 simp1 994 . . . 4  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  S  e.  W
)
3 simp2 995 . . . 4  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  I  e.  Fin )
4 eqid 2454 . . . 4  |-  ( dist `  Y )  =  (
dist `  Y )
5 eqid 2454 . . . 4  |-  ( Base `  Y )  =  (
Base `  Y )
6 simp3 996 . . . 4  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  R : I --> *MetSp )
71, 2, 3, 4, 5, 6prdsxmslem1 21197 . . 3  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  ( dist `  Y
)  e.  ( *Met `  ( Base `  Y ) ) )
8 ssid 3508 . . 3  |-  ( Base `  Y )  C_  ( Base `  Y )
9 xmetres2 21030 . . 3  |-  ( ( ( dist `  Y
)  e.  ( *Met `  ( Base `  Y ) )  /\  ( Base `  Y )  C_  ( Base `  Y
) )  ->  (
( dist `  Y )  |`  ( ( Base `  Y
)  X.  ( Base `  Y ) ) )  e.  ( *Met `  ( Base `  Y
) ) )
107, 8, 9sylancl 660 . 2  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  ( ( dist `  Y )  |`  (
( Base `  Y )  X.  ( Base `  Y
) ) )  e.  ( *Met `  ( Base `  Y )
) )
11 eqid 2454 . . . 4  |-  ( TopOpen `  Y )  =  (
TopOpen `  Y )
12 eqid 2454 . . . 4  |-  ( Base `  ( R `  k
) )  =  (
Base `  ( R `  k ) )
13 eqid 2454 . . . 4  |-  ( (
dist `  ( R `  k ) )  |`  ( ( Base `  ( R `  k )
)  X.  ( Base `  ( R `  k
) ) ) )  =  ( ( dist `  ( R `  k
) )  |`  (
( Base `  ( R `  k ) )  X.  ( Base `  ( R `  k )
) ) )
14 eqid 2454 . . . 4  |-  ( TopOpen `  ( R `  k ) )  =  ( TopOpen `  ( R `  k ) )
15 eqid 2454 . . . 4  |-  { x  |  E. g ( ( g  Fn  I  /\  A. k  e.  I  ( g `  k )  e.  ( ( TopOpen  o.  R ) `  k
)  /\  E. z  e.  Fin  A. k  e.  ( I  \  z
) ( g `  k )  =  U. ( ( TopOpen  o.  R
) `  k )
)  /\  x  =  X_ k  e.  I  ( g `  k ) ) }  =  {
x  |  E. g
( ( g  Fn  I  /\  A. k  e.  I  ( g `  k )  e.  ( ( TopOpen  o.  R ) `  k )  /\  E. z  e.  Fin  A. k  e.  ( I  \  z
) ( g `  k )  =  U. ( ( TopOpen  o.  R
) `  k )
)  /\  x  =  X_ k  e.  I  ( g `  k ) ) }
161, 2, 3, 4, 5, 6, 11, 12, 13, 14, 15prdsxmslem2 21198 . . 3  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  ( TopOpen `  Y
)  =  ( MetOpen `  ( dist `  Y )
) )
17 xmetf 20998 . . . . 5  |-  ( (
dist `  Y )  e.  ( *Met `  ( Base `  Y )
)  ->  ( dist `  Y ) : ( ( Base `  Y
)  X.  ( Base `  Y ) ) --> RR* )
18 ffn 5713 . . . . 5  |-  ( (
dist `  Y ) : ( ( Base `  Y )  X.  ( Base `  Y ) ) -->
RR*  ->  ( dist `  Y
)  Fn  ( (
Base `  Y )  X.  ( Base `  Y
) ) )
19 fnresdm 5672 . . . . 5  |-  ( (
dist `  Y )  Fn  ( ( Base `  Y
)  X.  ( Base `  Y ) )  -> 
( ( dist `  Y
)  |`  ( ( Base `  Y )  X.  ( Base `  Y ) ) )  =  ( dist `  Y ) )
207, 17, 18, 194syl 21 . . . 4  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  ( ( dist `  Y )  |`  (
( Base `  Y )  X.  ( Base `  Y
) ) )  =  ( dist `  Y
) )
2120fveq2d 5852 . . 3  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  ( MetOpen `  (
( dist `  Y )  |`  ( ( Base `  Y
)  X.  ( Base `  Y ) ) ) )  =  ( MetOpen `  ( dist `  Y )
) )
2216, 21eqtr4d 2498 . 2  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  ( TopOpen `  Y
)  =  ( MetOpen `  ( ( dist `  Y
)  |`  ( ( Base `  Y )  X.  ( Base `  Y ) ) ) ) )
23 eqid 2454 . . 3  |-  ( (
dist `  Y )  |`  ( ( Base `  Y
)  X.  ( Base `  Y ) ) )  =  ( ( dist `  Y )  |`  (
( Base `  Y )  X.  ( Base `  Y
) ) )
2411, 5, 23isxms2 21117 . 2  |-  ( Y  e.  *MetSp  <->  ( (
( dist `  Y )  |`  ( ( Base `  Y
)  X.  ( Base `  Y ) ) )  e.  ( *Met `  ( Base `  Y
) )  /\  ( TopOpen
`  Y )  =  ( MetOpen `  ( ( dist `  Y )  |`  ( ( Base `  Y
)  X.  ( Base `  Y ) ) ) ) ) )
2510, 22, 24sylanbrc 662 1  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> *MetSp )  ->  Y  e.  *MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805    \ cdif 3458    C_ wss 3461   U.cuni 4235    X. cxp 4986    |` cres 4990    o. ccom 4992    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   X_cixp 7462   Fincfn 7509   RR*cxr 9616   Basecbs 14716   distcds 14793   TopOpenctopn 14911   X_scprds 14935   *Metcxmt 18598   MetOpencmopn 18603   *MetSpcxme 20986
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-iin 4318  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-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  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-icc 11539  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-topgen 14933  df-pt 14934  df-prds 14937  df-psmet 18606  df-xmet 18607  df-bl 18609  df-mopn 18610  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-xms 20989
This theorem is referenced by:  prdsms  21200  pwsxms  21201  xpsxms  21203
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