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Theorem tmsxpsval 18084
Description: Value of the product of two metrics. (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 ) )
tmsxpsval.a  |-  ( ph  ->  A  e.  X )
tmsxpsval.b  |-  ( ph  ->  B  e.  Y )
tmsxpsval.c  |-  ( ph  ->  C  e.  X )
tmsxpsval.d  |-  ( ph  ->  D  e.  Y )
Assertion
Ref Expression
tmsxpsval  |-  ( ph  ->  ( <. A ,  B >. P <. C ,  D >. )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )

Proof of Theorem tmsxpsval
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
2 eqid 2283 . . 3  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
3 eqid 2283 . . 3  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
4 tmsxps.1 . . . 4  |-  ( ph  ->  M  e.  ( * Met `  X ) )
5 eqid 2283 . . . . 5  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
65tmsxms 18032 . . . 4  |-  ( M  e.  ( * Met `  X )  ->  (toMetSp `  M )  e.  * MetSp )
74, 6syl 15 . . 3  |-  ( ph  ->  (toMetSp `  M )  e.  * MetSp )
8 tmsxps.2 . . . 4  |-  ( ph  ->  N  e.  ( * Met `  Y ) )
9 eqid 2283 . . . . 5  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
109tmsxms 18032 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  (toMetSp `  N )  e.  * MetSp )
118, 10syl 15 . . 3  |-  ( ph  ->  (toMetSp `  N )  e.  * MetSp )
12 tmsxps.p . . 3  |-  P  =  ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )
13 eqid 2283 . . 3  |-  ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) )  =  ( ( dist `  (toMetSp `  M )
)  |`  ( ( Base `  (toMetSp `  M )
)  X.  ( Base `  (toMetSp `  M )
) ) )
14 eqid 2283 . . 3  |-  ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) )  =  ( ( dist `  (toMetSp `  N )
)  |`  ( ( Base `  (toMetSp `  N )
)  X.  ( Base `  (toMetSp `  N )
) ) )
155tmsds 18030 . . . . . . 7  |-  ( M  e.  ( * Met `  X )  ->  M  =  ( dist `  (toMetSp `  M ) ) )
164, 15syl 15 . . . . . 6  |-  ( ph  ->  M  =  ( dist `  (toMetSp `  M )
) )
175tmsbas 18029 . . . . . . . 8  |-  ( M  e.  ( * Met `  X )  ->  X  =  ( Base `  (toMetSp `  M ) ) )
184, 17syl 15 . . . . . . 7  |-  ( ph  ->  X  =  ( Base `  (toMetSp `  M )
) )
1918fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( * Met `  X
)  =  ( * Met `  ( Base `  (toMetSp `  M )
) ) )
2016, 19eleq12d 2351 . . . . 5  |-  ( ph  ->  ( M  e.  ( * Met `  X
)  <->  ( dist `  (toMetSp `  M ) )  e.  ( * Met `  ( Base `  (toMetSp `  M
) ) ) ) )
214, 20mpbid 201 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  M ) )  e.  ( * Met `  ( Base `  (toMetSp `  M
) ) ) )
22 ssid 3197 . . . 4  |-  ( Base `  (toMetSp `  M )
)  C_  ( Base `  (toMetSp `  M )
)
23 xmetres2 17925 . . . 4  |-  ( ( ( dist `  (toMetSp `  M ) )  e.  ( * Met `  ( Base `  (toMetSp `  M
) ) )  /\  ( Base `  (toMetSp `  M
) )  C_  ( Base `  (toMetSp `  M
) ) )  -> 
( ( dist `  (toMetSp `  M ) )  |`  ( ( Base `  (toMetSp `  M ) )  X.  ( Base `  (toMetSp `  M ) ) ) )  e.  ( * Met `  ( Base `  (toMetSp `  M )
) ) )
2421, 22, 23sylancl 643 . . 3  |-  ( ph  ->  ( ( dist `  (toMetSp `  M ) )  |`  ( ( Base `  (toMetSp `  M ) )  X.  ( Base `  (toMetSp `  M ) ) ) )  e.  ( * Met `  ( Base `  (toMetSp `  M )
) ) )
259tmsds 18030 . . . . . . 7  |-  ( N  e.  ( * Met `  Y )  ->  N  =  ( dist `  (toMetSp `  N ) ) )
268, 25syl 15 . . . . . 6  |-  ( ph  ->  N  =  ( dist `  (toMetSp `  N )
) )
279tmsbas 18029 . . . . . . . 8  |-  ( N  e.  ( * Met `  Y )  ->  Y  =  ( Base `  (toMetSp `  N ) ) )
288, 27syl 15 . . . . . . 7  |-  ( ph  ->  Y  =  ( Base `  (toMetSp `  N )
) )
2928fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( * Met `  Y
)  =  ( * Met `  ( Base `  (toMetSp `  N )
) ) )
3026, 29eleq12d 2351 . . . . 5  |-  ( ph  ->  ( N  e.  ( * Met `  Y
)  <->  ( dist `  (toMetSp `  N ) )  e.  ( * Met `  ( Base `  (toMetSp `  N
) ) ) ) )
318, 30mpbid 201 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  N ) )  e.  ( * Met `  ( Base `  (toMetSp `  N
) ) ) )
32 ssid 3197 . . . 4  |-  ( Base `  (toMetSp `  N )
)  C_  ( Base `  (toMetSp `  N )
)
33 xmetres2 17925 . . . 4  |-  ( ( ( dist `  (toMetSp `  N ) )  e.  ( * Met `  ( Base `  (toMetSp `  N
) ) )  /\  ( Base `  (toMetSp `  N
) )  C_  ( Base `  (toMetSp `  N
) ) )  -> 
( ( dist `  (toMetSp `  N ) )  |`  ( ( Base `  (toMetSp `  N ) )  X.  ( Base `  (toMetSp `  N ) ) ) )  e.  ( * Met `  ( Base `  (toMetSp `  N )
) ) )
3431, 32, 33sylancl 643 . . 3  |-  ( ph  ->  ( ( dist `  (toMetSp `  N ) )  |`  ( ( Base `  (toMetSp `  N ) )  X.  ( Base `  (toMetSp `  N ) ) ) )  e.  ( * Met `  ( Base `  (toMetSp `  N )
) ) )
35 tmsxpsval.a . . . 4  |-  ( ph  ->  A  e.  X )
3635, 18eleqtrd 2359 . . 3  |-  ( ph  ->  A  e.  ( Base `  (toMetSp `  M )
) )
37 tmsxpsval.b . . . 4  |-  ( ph  ->  B  e.  Y )
3837, 28eleqtrd 2359 . . 3  |-  ( ph  ->  B  e.  ( Base `  (toMetSp `  N )
) )
39 tmsxpsval.c . . . 4  |-  ( ph  ->  C  e.  X )
4039, 18eleqtrd 2359 . . 3  |-  ( ph  ->  C  e.  ( Base `  (toMetSp `  M )
) )
41 tmsxpsval.d . . . 4  |-  ( ph  ->  D  e.  Y )
4241, 28eleqtrd 2359 . . 3  |-  ( ph  ->  D  e.  ( Base `  (toMetSp `  N )
) )
431, 2, 3, 7, 11, 12, 13, 14, 24, 34, 36, 38, 40, 42xpsdsval 17945 . 2  |-  ( ph  ->  ( <. A ,  B >. P <. C ,  D >. )  =  sup ( { ( A ( ( dist `  (toMetSp `  M ) )  |`  ( ( Base `  (toMetSp `  M ) )  X.  ( Base `  (toMetSp `  M ) ) ) ) C ) ,  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D ) } ,  RR* ,  <  ) )
4436, 40ovresd 5988 . . . . 5  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4516oveqd 5875 . . . . 5  |-  ( ph  ->  ( A M C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4644, 45eqtr4d 2318 . . . 4  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A M C ) )
4738, 42ovresd 5988 . . . . 5  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4826oveqd 5875 . . . . 5  |-  ( ph  ->  ( B N D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4947, 48eqtr4d 2318 . . . 4  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B N D ) )
5046, 49preq12d 3714 . . 3  |-  ( ph  ->  { ( A ( ( dist `  (toMetSp `  M ) )  |`  ( ( Base `  (toMetSp `  M ) )  X.  ( Base `  (toMetSp `  M ) ) ) ) C ) ,  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D ) }  =  { ( A M C ) ,  ( B N D ) } )
5150supeq1d 7199 . 2  |-  ( ph  ->  sup ( { ( A ( ( dist `  (toMetSp `  M )
)  |`  ( ( Base `  (toMetSp `  M )
)  X.  ( Base `  (toMetSp `  M )
) ) ) C ) ,  ( B ( ( dist `  (toMetSp `  N ) )  |`  ( ( Base `  (toMetSp `  N ) )  X.  ( Base `  (toMetSp `  N ) ) ) ) D ) } ,  RR* ,  <  )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
5243, 51eqtrd 2315 1  |-  ( ph  ->  ( <. A ,  B >. P <. C ,  D >. )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152   {cpr 3641   <.cop 3643    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   supcsup 7193   RR*cxr 8866    < clt 8867   Basecbs 13148   distcds 13217    X.s cxps 13409   * Metcxmt 16369   * MetSpcxme 17882  toMetSpctmt 17884
This theorem is referenced by:  tmsxpsval2  18085
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-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-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  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-xms 17885  df-tms 17887
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