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Theorem tmsxpsmopn 18083
Description: Express the product of two metrics as another metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tmsxps.p  |-  P  =  ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )
tmsxps.1  |-  ( ph  ->  M  e.  ( * Met `  X ) )
tmsxps.2  |-  ( ph  ->  N  e.  ( * Met `  Y ) )
tmsxpsmopn.j  |-  J  =  ( MetOpen `  M )
tmsxpsmopn.k  |-  K  =  ( MetOpen `  N )
tmsxpsmopn.l  |-  L  =  ( MetOpen `  P )
Assertion
Ref Expression
tmsxpsmopn  |-  ( ph  ->  L  =  ( J 
tX  K ) )

Proof of Theorem tmsxpsmopn
StepHypRef Expression
1 tmsxps.1 . . . . 5  |-  ( ph  ->  M  e.  ( * Met `  X ) )
2 eqid 2283 . . . . . 6  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
32tmsxms 18032 . . . . 5  |-  ( M  e.  ( * Met `  X )  ->  (toMetSp `  M )  e.  * MetSp )
41, 3syl 15 . . . 4  |-  ( ph  ->  (toMetSp `  M )  e.  * MetSp )
5 xmstps 17999 . . . 4  |-  ( (toMetSp `  M )  e.  * MetSp  ->  (toMetSp `  M )  e.  TopSp )
64, 5syl 15 . . 3  |-  ( ph  ->  (toMetSp `  M )  e.  TopSp )
7 tmsxps.2 . . . . 5  |-  ( ph  ->  N  e.  ( * Met `  Y ) )
8 eqid 2283 . . . . . 6  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
98tmsxms 18032 . . . . 5  |-  ( N  e.  ( * Met `  Y )  ->  (toMetSp `  N )  e.  * MetSp )
107, 9syl 15 . . . 4  |-  ( ph  ->  (toMetSp `  N )  e.  * MetSp )
11 xmstps 17999 . . . 4  |-  ( (toMetSp `  N )  e.  * MetSp  ->  (toMetSp `  N )  e.  TopSp )
1210, 11syl 15 . . 3  |-  ( ph  ->  (toMetSp `  N )  e.  TopSp )
13 eqid 2283 . . . 4  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
14 eqid 2283 . . . 4  |-  ( TopOpen `  (toMetSp `  M ) )  =  ( TopOpen `  (toMetSp `  M ) )
15 eqid 2283 . . . 4  |-  ( TopOpen `  (toMetSp `  N ) )  =  ( TopOpen `  (toMetSp `  N ) )
16 eqid 2283 . . . 4  |-  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )
1713, 14, 15, 16xpstopn 17503 . . 3  |-  ( ( (toMetSp `  M )  e.  TopSp  /\  (toMetSp `  N
)  e.  TopSp )  -> 
( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  =  ( ( TopOpen `  (toMetSp `  M ) )  tX  ( TopOpen `  (toMetSp `  N
) ) ) )
186, 12, 17syl2anc 642 . 2  |-  ( ph  ->  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  =  ( ( TopOpen `  (toMetSp `  M ) )  tX  ( TopOpen `  (toMetSp `  N
) ) ) )
19 tmsxpsmopn.l . . 3  |-  L  =  ( MetOpen `  P )
2013xpsxms 18080 . . . . . 6  |-  ( ( (toMetSp `  M )  e.  * MetSp  /\  (toMetSp `  N )  e.  * MetSp )  ->  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  e.  * MetSp )
214, 10, 20syl2anc 642 . . . . 5  |-  ( ph  ->  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
)  e.  * MetSp )
22 eqid 2283 . . . . . 6  |-  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) )  =  (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )
23 tmsxps.p . . . . . . 7  |-  P  =  ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )
2423reseq1i 4951 . . . . . 6  |-  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  =  ( ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
2516, 22, 24xmstopn 17997 . . . . 5  |-  ( ( (toMetSp `  M )  X.s  (toMetSp `  N ) )  e.  * MetSp  ->  ( TopOpen
`  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) )  =  (
MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) ) )
2621, 25syl 15 . . . 4  |-  ( ph  ->  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  =  ( MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) ) )
27 eqid 2283 . . . . . . 7  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
28 eqid 2283 . . . . . . 7  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
2913, 27, 28, 4, 10, 23xpsdsfn2 17942 . . . . . 6  |-  ( ph  ->  P  Fn  ( (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
30 fnresdm 5353 . . . . . 6  |-  ( P  Fn  ( ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) )  X.  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) ) )  -> 
( P  |`  (
( Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  =  P )
3129, 30syl 15 . . . . 5  |-  ( ph  ->  ( P  |`  (
( Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  =  P )
3231fveq2d 5529 . . . 4  |-  ( ph  ->  ( MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) )  =  ( MetOpen `  P )
)
3326, 32eqtr2d 2316 . . 3  |-  ( ph  ->  ( MetOpen `  P )  =  ( TopOpen `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
3419, 33syl5eq 2327 . 2  |-  ( ph  ->  L  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
35 tmsxpsmopn.j . . . . 5  |-  J  =  ( MetOpen `  M )
362, 35tmstopn 18031 . . . 4  |-  ( M  e.  ( * Met `  X )  ->  J  =  ( TopOpen `  (toMetSp `  M ) ) )
371, 36syl 15 . . 3  |-  ( ph  ->  J  =  ( TopOpen `  (toMetSp `  M ) ) )
38 tmsxpsmopn.k . . . . 5  |-  K  =  ( MetOpen `  N )
398, 38tmstopn 18031 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
407, 39syl 15 . . 3  |-  ( ph  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
4137, 40oveq12d 5876 . 2  |-  ( ph  ->  ( J  tX  K
)  =  ( (
TopOpen `  (toMetSp `  M
) )  tX  ( TopOpen
`  (toMetSp `  N )
) ) )
4218, 34, 413eqtr4d 2325 1  |-  ( ph  ->  L  =  ( J 
tX  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    X. cxp 4687    |` cres 4691    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Basecbs 13148   distcds 13217   TopOpenctopn 13326    X.s cxps 13409   * Metcxmt 16369   MetOpencmopn 16372   TopSpctps 16634    tX ctx 17255   *
MetSpcxme 17882  toMetSpctmt 17884
This theorem is referenced by:  txmetcnp  18093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-xms 17885  df-tms 17887
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