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Theorem tmsxps 18098
Description: Express the product of two metrics as another metric. (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 ) )
Assertion
Ref Expression
tmsxps  |-  ( ph  ->  P  e.  ( * Met `  ( X  X.  Y ) ) )

Proof of Theorem tmsxps
StepHypRef Expression
1 eqid 2296 . . . . 5  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
2 eqid 2296 . . . . 5  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
3 eqid 2296 . . . . 5  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
4 tmsxps.1 . . . . . 6  |-  ( ph  ->  M  e.  ( * Met `  X ) )
5 eqid 2296 . . . . . . 7  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
65tmsxms 18048 . . . . . 6  |-  ( M  e.  ( * Met `  X )  ->  (toMetSp `  M )  e.  * MetSp )
74, 6syl 15 . . . . 5  |-  ( ph  ->  (toMetSp `  M )  e.  * MetSp )
8 tmsxps.2 . . . . . 6  |-  ( ph  ->  N  e.  ( * Met `  Y ) )
9 eqid 2296 . . . . . . 7  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
109tmsxms 18048 . . . . . 6  |-  ( N  e.  ( * Met `  Y )  ->  (toMetSp `  N )  e.  * MetSp )
118, 10syl 15 . . . . 5  |-  ( ph  ->  (toMetSp `  N )  e.  * MetSp )
12 tmsxps.p . . . . 5  |-  P  =  ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )
131, 2, 3, 7, 11, 12xpsdsfn2 17958 . . . 4  |-  ( ph  ->  P  Fn  ( (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
14 fnresdm 5369 . . . 4  |-  ( P  Fn  ( ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) )  X.  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) ) )  -> 
( P  |`  (
( Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  =  P )
1513, 14syl 15 . . 3  |-  ( ph  ->  ( P  |`  (
( Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  =  P )
161xpsxms 18096 . . . . 5  |-  ( ( (toMetSp `  M )  e.  * MetSp  /\  (toMetSp `  N )  e.  * MetSp )  ->  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  e.  * MetSp )
177, 11, 16syl2anc 642 . . . 4  |-  ( ph  ->  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
)  e.  * MetSp )
18 eqid 2296 . . . . 5  |-  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) )  =  (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )
1918, 12xmsxmet2 18021 . . . 4  |-  ( ( (toMetSp `  M )  X.s  (toMetSp `  N ) )  e.  * MetSp  ->  ( P  |`  ( ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) )  X.  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) ) ) )  e.  ( * Met `  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
2017, 19syl 15 . . 3  |-  ( ph  ->  ( P  |`  (
( Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  e.  ( * Met `  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) ) ) )
2115, 20eqeltrrd 2371 . 2  |-  ( ph  ->  P  e.  ( * Met `  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) ) ) )
225tmsbas 18045 . . . . . 6  |-  ( M  e.  ( * Met `  X )  ->  X  =  ( Base `  (toMetSp `  M ) ) )
234, 22syl 15 . . . . 5  |-  ( ph  ->  X  =  ( Base `  (toMetSp `  M )
) )
249tmsbas 18045 . . . . . 6  |-  ( N  e.  ( * Met `  Y )  ->  Y  =  ( Base `  (toMetSp `  N ) ) )
258, 24syl 15 . . . . 5  |-  ( ph  ->  Y  =  ( Base `  (toMetSp `  N )
) )
2623, 25xpeq12d 4730 . . . 4  |-  ( ph  ->  ( X  X.  Y
)  =  ( (
Base `  (toMetSp `  M
) )  X.  ( Base `  (toMetSp `  N
) ) ) )
271, 2, 3, 7, 11xpsbas 13492 . . . 4  |-  ( ph  ->  ( ( Base `  (toMetSp `  M ) )  X.  ( Base `  (toMetSp `  N ) ) )  =  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
2826, 27eqtrd 2328 . . 3  |-  ( ph  ->  ( X  X.  Y
)  =  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) ) )
2928fveq2d 5545 . 2  |-  ( ph  ->  ( * Met `  ( X  X.  Y ) )  =  ( * Met `  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
3021, 29eleqtrrd 2373 1  |-  ( ph  ->  P  e.  ( * Met `  ( X  X.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    X. cxp 4703    |` cres 4707    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   Basecbs 13164   distcds 13233    X.s cxps 13425   * Metcxmt 16385   * MetSpcxme 17898  toMetSpctmt 17900
This theorem is referenced by:  txmetcnp  18109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901  df-tms 17903
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