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Theorem tmsxpsval 21484
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 2429 . . 3  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
2 eqid 2429 . . 3  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
3 eqid 2429 . . 3  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
4 tmsxps.1 . . . 4  |-  ( ph  ->  M  e.  ( *Met `  X ) )
5 eqid 2429 . . . . 5  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
65tmsxms 21432 . . . 4  |-  ( M  e.  ( *Met `  X )  ->  (toMetSp `  M )  e.  *MetSp )
74, 6syl 17 . . 3  |-  ( ph  ->  (toMetSp `  M )  e.  *MetSp )
8 tmsxps.2 . . . 4  |-  ( ph  ->  N  e.  ( *Met `  Y ) )
9 eqid 2429 . . . . 5  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
109tmsxms 21432 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  (toMetSp `  N )  e.  *MetSp )
118, 10syl 17 . . 3  |-  ( ph  ->  (toMetSp `  N )  e.  *MetSp )
12 tmsxps.p . . 3  |-  P  =  ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )
13 eqid 2429 . . 3  |-  ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) )  =  ( ( dist `  (toMetSp `  M )
)  |`  ( ( Base `  (toMetSp `  M )
)  X.  ( Base `  (toMetSp `  M )
) ) )
14 eqid 2429 . . 3  |-  ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) )  =  ( ( dist `  (toMetSp `  N )
)  |`  ( ( Base `  (toMetSp `  N )
)  X.  ( Base `  (toMetSp `  N )
) ) )
155tmsds 21430 . . . . . 6  |-  ( M  e.  ( *Met `  X )  ->  M  =  ( dist `  (toMetSp `  M ) ) )
164, 15syl 17 . . . . 5  |-  ( ph  ->  M  =  ( dist `  (toMetSp `  M )
) )
175tmsbas 21429 . . . . . . 7  |-  ( M  e.  ( *Met `  X )  ->  X  =  ( Base `  (toMetSp `  M ) ) )
184, 17syl 17 . . . . . 6  |-  ( ph  ->  X  =  ( Base `  (toMetSp `  M )
) )
1918fveq2d 5885 . . . . 5  |-  ( ph  ->  ( *Met `  X )  =  ( *Met `  ( Base `  (toMetSp `  M
) ) ) )
204, 16, 193eltr3d 2531 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  M ) )  e.  ( *Met `  ( Base `  (toMetSp `  M
) ) ) )
21 ssid 3489 . . . 4  |-  ( Base `  (toMetSp `  M )
)  C_  ( Base `  (toMetSp `  M )
)
22 xmetres2 21307 . . . 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 666 . . 3  |-  ( ph  ->  ( ( dist `  (toMetSp `  M ) )  |`  ( ( Base `  (toMetSp `  M ) )  X.  ( Base `  (toMetSp `  M ) ) ) )  e.  ( *Met `  ( Base `  (toMetSp `  M )
) ) )
249tmsds 21430 . . . . . 6  |-  ( N  e.  ( *Met `  Y )  ->  N  =  ( dist `  (toMetSp `  N ) ) )
258, 24syl 17 . . . . 5  |-  ( ph  ->  N  =  ( dist `  (toMetSp `  N )
) )
269tmsbas 21429 . . . . . . 7  |-  ( N  e.  ( *Met `  Y )  ->  Y  =  ( Base `  (toMetSp `  N ) ) )
278, 26syl 17 . . . . . 6  |-  ( ph  ->  Y  =  ( Base `  (toMetSp `  N )
) )
2827fveq2d 5885 . . . . 5  |-  ( ph  ->  ( *Met `  Y )  =  ( *Met `  ( Base `  (toMetSp `  N
) ) ) )
298, 25, 283eltr3d 2531 . . . 4  |-  ( ph  ->  ( dist `  (toMetSp `  N ) )  e.  ( *Met `  ( Base `  (toMetSp `  N
) ) ) )
30 ssid 3489 . . . 4  |-  ( Base `  (toMetSp `  N )
)  C_  ( Base `  (toMetSp `  N )
)
31 xmetres2 21307 . . . 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 666 . . 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 21327 . 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 6451 . . . . 5  |-  ( ph  ->  ( A ( (
dist `  (toMetSp `  M
) )  |`  (
( Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  M
) ) ) ) C )  =  ( A ( dist `  (toMetSp `  M ) ) C ) )
4316oveqd 6322 . . . . 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 6451 . . . . 5  |-  ( ph  ->  ( B ( (
dist `  (toMetSp `  N
) )  |`  (
( Base `  (toMetSp `  N
) )  X.  ( Base `  (toMetSp `  N
) ) ) ) D )  =  ( B ( dist `  (toMetSp `  N ) ) D ) )
4625oveqd 6322 . . . . 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 4090 . . 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 7966 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    C_ wss 3442   {cpr 4004   <.cop 4008    X. cxp 4852    |` cres 4856   ` cfv 5601  (class class class)co 6305   supcsup 7960   RR*cxr 9673    < clt 9674   Basecbs 15084   distcds 15161    X.s cxps 15363   *Metcxmt 18890   *MetSpcxme 21263  toMetSpctmt 21265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-prds 15305  df-xrs 15359  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-bl 18900  df-mopn 18901  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-xms 21266  df-tms 21268
This theorem is referenced by:  tmsxpsval2  21485
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