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Theorem tmsxpsval 20113
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 2443 . . 3  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
2 eqid 2443 . . 3  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
3 eqid 2443 . . 3  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
4 tmsxps.1 . . . 4  |-  ( ph  ->  M  e.  ( *Met `  X ) )
5 eqid 2443 . . . . 5  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
65tmsxms 20061 . . . 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 2443 . . . . 5  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
109tmsxms 20061 . . . 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 2443 . . 3  |-  ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) )  =  ( ( dist `  (toMetSp `  M )
)  |`  ( ( Base `  (toMetSp `  M )
)  X.  ( Base `  (toMetSp `  M )
) ) )
14 eqid 2443 . . 3  |-  ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) )  =  ( ( dist `  (toMetSp `  N )
)  |`  ( ( Base `  (toMetSp `  N )
)  X.  ( Base `  (toMetSp `  N )
) ) )
155tmsds 20059 . . . . . 6  |-  ( M  e.  ( *Met `  X )  ->  M  =  ( dist `  (toMetSp `  M ) ) )
164, 15syl 16 . . . . 5  |-  ( ph  ->  M  =  ( dist `  (toMetSp `  M )
) )
175tmsbas 20058 . . . . . . 7  |-  ( M  e.  ( *Met `  X )  ->  X  =  ( Base `  (toMetSp `  M ) ) )
184, 17syl 16 . . . . . 6  |-  ( ph  ->  X  =  ( Base `  (toMetSp `  M )
) )
1918fveq2d 5695 . . . . 5  |-  ( ph  ->  ( *Met `  X )  =  ( *Met `  ( Base `  (toMetSp `  M
) ) ) )
204, 16, 193eltr3d 2523 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  M ) )  e.  ( *Met `  ( Base `  (toMetSp `  M
) ) ) )
21 ssid 3375 . . . 4  |-  ( Base `  (toMetSp `  M )
)  C_  ( Base `  (toMetSp `  M )
)
22 xmetres2 19936 . . . 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 20059 . . . . . 6  |-  ( N  e.  ( *Met `  Y )  ->  N  =  ( dist `  (toMetSp `  N ) ) )
258, 24syl 16 . . . . 5  |-  ( ph  ->  N  =  ( dist `  (toMetSp `  N )
) )
269tmsbas 20058 . . . . . . 7  |-  ( N  e.  ( *Met `  Y )  ->  Y  =  ( Base `  (toMetSp `  N ) ) )
278, 26syl 16 . . . . . 6  |-  ( ph  ->  Y  =  ( Base `  (toMetSp `  N )
) )
2827fveq2d 5695 . . . . 5  |-  ( ph  ->  ( *Met `  Y )  =  ( *Met `  ( Base `  (toMetSp `  N
) ) ) )
298, 25, 283eltr3d 2523 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  N ) )  e.  ( *Met `  ( Base `  (toMetSp `  N
) ) ) )
30 ssid 3375 . . . 4  |-  ( Base `  (toMetSp `  N )
)  C_  ( Base `  (toMetSp `  N )
)
31 xmetres2 19936 . . . 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 2519 . . 3  |-  ( ph  ->  A  e.  ( Base `  (toMetSp `  M )
) )
35 tmsxpsval.b . . . 4  |-  ( ph  ->  B  e.  Y )
3635, 27eleqtrd 2519 . . 3  |-  ( ph  ->  B  e.  ( Base `  (toMetSp `  N )
) )
37 tmsxpsval.c . . . 4  |-  ( ph  ->  C  e.  X )
3837, 18eleqtrd 2519 . . 3  |-  ( ph  ->  C  e.  ( Base `  (toMetSp `  M )
) )
39 tmsxpsval.d . . . 4  |-  ( ph  ->  D  e.  Y )
4039, 27eleqtrd 2519 . . 3  |-  ( ph  ->  D  e.  ( Base `  (toMetSp `  N )
) )
411, 2, 3, 7, 11, 12, 13, 14, 23, 32, 34, 36, 38, 40xpsdsval 19956 . 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 6231 . . . . 5  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4316oveqd 6108 . . . . 5  |-  ( ph  ->  ( A M C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4442, 43eqtr4d 2478 . . . 4  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A M C ) )
4536, 40ovresd 6231 . . . . 5  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4625oveqd 6108 . . . . 5  |-  ( ph  ->  ( B N D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4745, 46eqtr4d 2478 . . . 4  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B N D ) )
4844, 47preq12d 3962 . . 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 7696 . 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 2475 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 1369    e. wcel 1756    C_ wss 3328   {cpr 3879   <.cop 3883    X. cxp 4838    |` cres 4842   ` cfv 5418  (class class class)co 6091   supcsup 7690   RR*cxr 9417    < clt 9418   Basecbs 14174   distcds 14247    X.s cxps 14444   *Metcxmt 17801   *MetSpcxme 19892  toMetSpctmt 19894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-prds 14386  df-xrs 14440  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-bl 17812  df-mopn 17813  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-xms 19895  df-tms 19897
This theorem is referenced by:  tmsxpsval2  20114
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