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Theorem tmsxpsval 20868
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 2467 . . 3  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
2 eqid 2467 . . 3  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
3 eqid 2467 . . 3  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
4 tmsxps.1 . . . 4  |-  ( ph  ->  M  e.  ( *Met `  X ) )
5 eqid 2467 . . . . 5  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
65tmsxms 20816 . . . 4  |-  ( M  e.  ( *Met `  X )  ->  (toMetSp `  M )  e.  *MetSp )
74, 6syl 16 . . 3  |-  ( ph  ->  (toMetSp `  M )  e.  *MetSp )
8 tmsxps.2 . . . 4  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
9 eqid 2467 . . . . 5  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
109tmsxms 20816 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  (toMetSp `  N )  e.  *MetSp )
118, 10syl 16 . . 3  |-  ( ph  ->  (toMetSp `  N )  e.  *MetSp )
12 tmsxps.p . . 3  |-  P  =  ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )
13 eqid 2467 . . 3  |-  ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) )  =  ( ( dist `  (toMetSp `  M )
)  |`  ( ( Base `  (toMetSp `  M )
)  X.  ( Base `  (toMetSp `  M )
) ) )
14 eqid 2467 . . 3  |-  ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) )  =  ( ( dist `  (toMetSp `  N )
)  |`  ( ( Base `  (toMetSp `  N )
)  X.  ( Base `  (toMetSp `  N )
) ) )
155tmsds 20814 . . . . . 6  |-  ( M  e.  ( *Met `  X )  ->  M  =  ( dist `  (toMetSp `  M ) ) )
164, 15syl 16 . . . . 5  |-  ( ph  ->  M  =  ( dist `  (toMetSp `  M )
) )
175tmsbas 20813 . . . . . . 7  |-  ( M  e.  ( *Met `  X )  ->  X  =  ( Base `  (toMetSp `  M ) ) )
184, 17syl 16 . . . . . 6  |-  ( ph  ->  X  =  ( Base `  (toMetSp `  M )
) )
1918fveq2d 5870 . . . . 5  |-  ( ph  ->  ( *Met `  X )  =  ( *Met `  ( Base `  (toMetSp `  M
) ) ) )
204, 16, 193eltr3d 2569 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  M ) )  e.  ( *Met `  ( Base `  (toMetSp `  M
) ) ) )
21 ssid 3523 . . . 4  |-  ( Base `  (toMetSp `  M )
)  C_  ( Base `  (toMetSp `  M )
)
22 xmetres2 20691 . . . 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 )
) ) )
2320, 21, 22sylancl 662 . . 3  |-  ( ph  ->  ( ( dist `  (toMetSp `  M ) )  |`  ( ( Base `  (toMetSp `  M ) )  X.  ( Base `  (toMetSp `  M ) ) ) )  e.  ( *Met `  ( Base `  (toMetSp `  M )
) ) )
249tmsds 20814 . . . . . 6  |-  ( N  e.  ( *Met `  Y )  ->  N  =  ( dist `  (toMetSp `  N ) ) )
258, 24syl 16 . . . . 5  |-  ( ph  ->  N  =  ( dist `  (toMetSp `  N )
) )
269tmsbas 20813 . . . . . . 7  |-  ( N  e.  ( *Met `  Y )  ->  Y  =  ( Base `  (toMetSp `  N ) ) )
278, 26syl 16 . . . . . 6  |-  ( ph  ->  Y  =  ( Base `  (toMetSp `  N )
) )
2827fveq2d 5870 . . . . 5  |-  ( ph  ->  ( *Met `  Y )  =  ( *Met `  ( Base `  (toMetSp `  N
) ) ) )
298, 25, 283eltr3d 2569 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  N ) )  e.  ( *Met `  ( Base `  (toMetSp `  N
) ) ) )
30 ssid 3523 . . . 4  |-  ( Base `  (toMetSp `  N )
)  C_  ( Base `  (toMetSp `  N )
)
31 xmetres2 20691 . . . 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 )
) ) )
3229, 30, 31sylancl 662 . . 3  |-  ( ph  ->  ( ( dist `  (toMetSp `  N ) )  |`  ( ( Base `  (toMetSp `  N ) )  X.  ( Base `  (toMetSp `  N ) ) ) )  e.  ( *Met `  ( Base `  (toMetSp `  N )
) ) )
33 tmsxpsval.a . . . 4  |-  ( ph  ->  A  e.  X )
3433, 18eleqtrd 2557 . . 3  |-  ( ph  ->  A  e.  ( Base `  (toMetSp `  M )
) )
35 tmsxpsval.b . . . 4  |-  ( ph  ->  B  e.  Y )
3635, 27eleqtrd 2557 . . 3  |-  ( ph  ->  B  e.  ( Base `  (toMetSp `  N )
) )
37 tmsxpsval.c . . . 4  |-  ( ph  ->  C  e.  X )
3837, 18eleqtrd 2557 . . 3  |-  ( ph  ->  C  e.  ( Base `  (toMetSp `  M )
) )
39 tmsxpsval.d . . . 4  |-  ( ph  ->  D  e.  Y )
4039, 27eleqtrd 2557 . . 3  |-  ( ph  ->  D  e.  ( Base `  (toMetSp `  N )
) )
411, 2, 3, 7, 11, 12, 13, 14, 23, 32, 34, 36, 38, 40xpsdsval 20711 . 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* ,  <  ) )
4234, 38ovresd 6428 . . . . 5  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4316oveqd 6302 . . . . 5  |-  ( ph  ->  ( A M C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4442, 43eqtr4d 2511 . . . 4  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A M C ) )
4536, 40ovresd 6428 . . . . 5  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4625oveqd 6302 . . . . 5  |-  ( ph  ->  ( B N D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4745, 46eqtr4d 2511 . . . 4  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B N D ) )
4844, 47preq12d 4114 . . 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 ) } )
4948supeq1d 7907 . 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* ,  <  ) )
5041, 49eqtrd 2508 1  |-  ( ph  ->  ( <. A ,  B >. P <. C ,  D >. )  =  sup ( { ( A M C ) ,  ( B N D ) } ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    C_ wss 3476   {cpr 4029   <.cop 4033    X. cxp 4997    |` cres 5001   ` cfv 5588  (class class class)co 6285   supcsup 7901   RR*cxr 9628    < clt 9629   Basecbs 14493   distcds 14567    X.s cxps 14764   *Metcxmt 18214   *MetSpcxme 20647  toMetSpctmt 20649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-sup 7902  df-oi 7936  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-icc 11537  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-hom 14582  df-cco 14583  df-rest 14681  df-topn 14682  df-0g 14700  df-gsum 14701  df-topgen 14702  df-prds 14706  df-xrs 14760  df-imas 14766  df-xps 14768  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-submnd 15790  df-mulg 15874  df-cntz 16169  df-cmn 16615  df-psmet 18222  df-xmet 18223  df-bl 18225  df-mopn 18226  df-top 19206  df-bases 19208  df-topon 19209  df-topsp 19210  df-xms 20650  df-tms 20652
This theorem is referenced by:  tmsxpsval2  20869
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