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Theorem tmsxpsval 18570
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 2438 . . 3  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
2 eqid 2438 . . 3  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
3 eqid 2438 . . 3  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
4 tmsxps.1 . . . 4  |-  ( ph  ->  M  e.  ( * Met `  X ) )
5 eqid 2438 . . . . 5  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
65tmsxms 18518 . . . 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 2438 . . . . 5  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
109tmsxms 18518 . . . 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 2438 . . 3  |-  ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) )  =  ( ( dist `  (toMetSp `  M )
)  |`  ( ( Base `  (toMetSp `  M )
)  X.  ( Base `  (toMetSp `  M )
) ) )
14 eqid 2438 . . 3  |-  ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) )  =  ( ( dist `  (toMetSp `  N )
)  |`  ( ( Base `  (toMetSp `  N )
)  X.  ( Base `  (toMetSp `  N )
) ) )
155tmsds 18516 . . . . . 6  |-  ( M  e.  ( * Met `  X )  ->  M  =  ( dist `  (toMetSp `  M ) ) )
164, 15syl 16 . . . . 5  |-  ( ph  ->  M  =  ( dist `  (toMetSp `  M )
) )
175tmsbas 18515 . . . . . . 7  |-  ( M  e.  ( * Met `  X )  ->  X  =  ( Base `  (toMetSp `  M ) ) )
184, 17syl 16 . . . . . 6  |-  ( ph  ->  X  =  ( Base `  (toMetSp `  M )
) )
1918fveq2d 5734 . . . . 5  |-  ( ph  ->  ( * Met `  X
)  =  ( * Met `  ( Base `  (toMetSp `  M )
) ) )
204, 16, 193eltr3d 2518 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  M ) )  e.  ( * Met `  ( Base `  (toMetSp `  M
) ) ) )
21 ssid 3369 . . . 4  |-  ( Base `  (toMetSp `  M )
)  C_  ( Base `  (toMetSp `  M )
)
22 xmetres2 18393 . . . 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 645 . . 3  |-  ( ph  ->  ( ( dist `  (toMetSp `  M ) )  |`  ( ( Base `  (toMetSp `  M ) )  X.  ( Base `  (toMetSp `  M ) ) ) )  e.  ( * Met `  ( Base `  (toMetSp `  M )
) ) )
249tmsds 18516 . . . . . 6  |-  ( N  e.  ( * Met `  Y )  ->  N  =  ( dist `  (toMetSp `  N ) ) )
258, 24syl 16 . . . . 5  |-  ( ph  ->  N  =  ( dist `  (toMetSp `  N )
) )
269tmsbas 18515 . . . . . . 7  |-  ( N  e.  ( * Met `  Y )  ->  Y  =  ( Base `  (toMetSp `  N ) ) )
278, 26syl 16 . . . . . 6  |-  ( ph  ->  Y  =  ( Base `  (toMetSp `  N )
) )
2827fveq2d 5734 . . . . 5  |-  ( ph  ->  ( * Met `  Y
)  =  ( * Met `  ( Base `  (toMetSp `  N )
) ) )
298, 25, 283eltr3d 2518 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  N ) )  e.  ( * Met `  ( Base `  (toMetSp `  N
) ) ) )
30 ssid 3369 . . . 4  |-  ( Base `  (toMetSp `  N )
)  C_  ( Base `  (toMetSp `  N )
)
31 xmetres2 18393 . . . 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 645 . . 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 2514 . . 3  |-  ( ph  ->  A  e.  ( Base `  (toMetSp `  M )
) )
35 tmsxpsval.b . . . 4  |-  ( ph  ->  B  e.  Y )
3635, 27eleqtrd 2514 . . 3  |-  ( ph  ->  B  e.  ( Base `  (toMetSp `  N )
) )
37 tmsxpsval.c . . . 4  |-  ( ph  ->  C  e.  X )
3837, 18eleqtrd 2514 . . 3  |-  ( ph  ->  C  e.  ( Base `  (toMetSp `  M )
) )
39 tmsxpsval.d . . . 4  |-  ( ph  ->  D  e.  Y )
4039, 27eleqtrd 2514 . . 3  |-  ( ph  ->  D  e.  ( Base `  (toMetSp `  N )
) )
411, 2, 3, 7, 11, 12, 13, 14, 23, 32, 34, 36, 38, 40xpsdsval 18413 . 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 6216 . . . . 5  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4316oveqd 6100 . . . . 5  |-  ( ph  ->  ( A M C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4442, 43eqtr4d 2473 . . . 4  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A M C ) )
4536, 40ovresd 6216 . . . . 5  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4625oveqd 6100 . . . . 5  |-  ( ph  ->  ( B N D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4745, 46eqtr4d 2473 . . . 4  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B N D ) )
4844, 47preq12d 3893 . . 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 7453 . 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 2470 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 1653    e. wcel 1726    C_ wss 3322   {cpr 3817   <.cop 3819    X. cxp 4878    |` cres 4882   ` cfv 5456  (class class class)co 6083   supcsup 7447   RR*cxr 9121    < clt 9122   Basecbs 13471   distcds 13540    X.s cxps 13734   * Metcxmt 16688   * MetSpcxme 18349  toMetSpctmt 18351
This theorem is referenced by:  tmsxpsval2  18571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-hash 11621  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-bl 16699  df-mopn 16700  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-xms 18352  df-tms 18354
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