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Theorem prdsxms 18128
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 955 . . . 4  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  S  e.  W
)
3 simp2 956 . . . 4  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  I  e.  Fin )
4 eqid 2316 . . . 4  |-  ( dist `  Y )  =  (
dist `  Y )
5 eqid 2316 . . . 4  |-  ( Base `  Y )  =  (
Base `  Y )
6 simp3 957 . . . 4  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  R : I --> * MetSp )
71, 2, 3, 4, 5, 6prdsxmslem1 18126 . . 3  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  ( dist `  Y
)  e.  ( * Met `  ( Base `  Y ) ) )
8 ssid 3231 . . 3  |-  ( Base `  Y )  C_  ( Base `  Y )
9 xmetres2 17977 . . 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 643 . 2  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  ( ( dist `  Y )  |`  (
( Base `  Y )  X.  ( Base `  Y
) ) )  e.  ( * Met `  ( Base `  Y ) ) )
11 eqid 2316 . . . 4  |-  ( TopOpen `  Y )  =  (
TopOpen `  Y )
12 eqid 2316 . . . 4  |-  ( Base `  ( R `  k
) )  =  (
Base `  ( R `  k ) )
13 eqid 2316 . . . 4  |-  ( (
dist `  ( R `  k ) )  |`  ( ( Base `  ( R `  k )
)  X.  ( Base `  ( R `  k
) ) ) )  =  ( ( dist `  ( R `  k
) )  |`  (
( Base `  ( R `  k ) )  X.  ( Base `  ( R `  k )
) ) )
14 eqid 2316 . . . 4  |-  ( TopOpen `  ( R `  k ) )  =  ( TopOpen `  ( R `  k ) )
15 eqid 2316 . . . 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 18127 . . 3  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  ( TopOpen `  Y
)  =  ( MetOpen `  ( dist `  Y )
) )
17 xmetf 17946 . . . . . 6  |-  ( (
dist `  Y )  e.  ( * Met `  ( Base `  Y ) )  ->  ( dist `  Y
) : ( (
Base `  Y )  X.  ( Base `  Y
) ) --> RR* )
187, 17syl 15 . . . . 5  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  ( dist `  Y
) : ( (
Base `  Y )  X.  ( Base `  Y
) ) --> RR* )
19 ffn 5427 . . . . 5  |-  ( (
dist `  Y ) : ( ( Base `  Y )  X.  ( Base `  Y ) ) -->
RR*  ->  ( dist `  Y
)  Fn  ( (
Base `  Y )  X.  ( Base `  Y
) ) )
20 fnresdm 5390 . . . . 5  |-  ( (
dist `  Y )  Fn  ( ( Base `  Y
)  X.  ( Base `  Y ) )  -> 
( ( dist `  Y
)  |`  ( ( Base `  Y )  X.  ( Base `  Y ) ) )  =  ( dist `  Y ) )
2118, 19, 203syl 18 . . . 4  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  ( ( dist `  Y )  |`  (
( Base `  Y )  X.  ( Base `  Y
) ) )  =  ( dist `  Y
) )
2221fveq2d 5567 . . 3  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  ( MetOpen `  (
( dist `  Y )  |`  ( ( Base `  Y
)  X.  ( Base `  Y ) ) ) )  =  ( MetOpen `  ( dist `  Y )
) )
2316, 22eqtr4d 2351 . 2  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  ( TopOpen `  Y
)  =  ( MetOpen `  ( ( dist `  Y
)  |`  ( ( Base `  Y )  X.  ( Base `  Y ) ) ) ) )
24 eqid 2316 . . 3  |-  ( (
dist `  Y )  |`  ( ( Base `  Y
)  X.  ( Base `  Y ) ) )  =  ( ( dist `  Y )  |`  (
( Base `  Y )  X.  ( Base `  Y
) ) )
2511, 5, 24isxms2 18046 . 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 ) ) ) ) ) )
2610, 23, 25sylanbrc 645 1  |-  ( ( S  e.  W  /\  I  e.  Fin  /\  R : I --> * MetSp )  ->  Y  e.  * MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1532    = wceq 1633    e. wcel 1701   {cab 2302   A.wral 2577   E.wrex 2578    \ cdif 3183    C_ wss 3186   U.cuni 3864    X. cxp 4724    |` cres 4728    o. ccom 4730    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900   X_cixp 6860   Fincfn 6906   RR*cxr 8911   Basecbs 13195   distcds 13264   TopOpenctopn 13375   X_scprds 13395   * Metcxmt 16418   MetOpencmopn 16423   *
MetSpcxme 17934
This theorem is referenced by:  prdsms  18129  pwsxms  18130  xpsxms  18132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-icc 10710  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-plusg 13268  df-mulr 13269  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xmet 16425  df-bl 16427  df-mopn 16428  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-xms 17937
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