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Theorem tmsxpsmopn 18524
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 ) )
tmsxpsmopn.j  |-  J  =  ( MetOpen `  M )
tmsxpsmopn.k  |-  K  =  ( MetOpen `  N )
tmsxpsmopn.l  |-  L  =  ( MetOpen `  P )
Assertion
Ref Expression
tmsxpsmopn  |-  ( ph  ->  L  =  ( J 
tX  K ) )

Proof of Theorem tmsxpsmopn
StepHypRef Expression
1 tmsxps.1 . . . . 5  |-  ( ph  ->  M  e.  ( * Met `  X ) )
2 eqid 2408 . . . . . 6  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
32tmsxms 18473 . . . . 5  |-  ( M  e.  ( * Met `  X )  ->  (toMetSp `  M )  e.  * MetSp )
41, 3syl 16 . . . 4  |-  ( ph  ->  (toMetSp `  M )  e.  * MetSp )
5 xmstps 18440 . . . 4  |-  ( (toMetSp `  M )  e.  * MetSp  ->  (toMetSp `  M )  e.  TopSp )
64, 5syl 16 . . 3  |-  ( ph  ->  (toMetSp `  M )  e.  TopSp )
7 tmsxps.2 . . . . 5  |-  ( ph  ->  N  e.  ( * Met `  Y ) )
8 eqid 2408 . . . . . 6  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
98tmsxms 18473 . . . . 5  |-  ( N  e.  ( * Met `  Y )  ->  (toMetSp `  N )  e.  * MetSp )
107, 9syl 16 . . . 4  |-  ( ph  ->  (toMetSp `  N )  e.  * MetSp )
11 xmstps 18440 . . . 4  |-  ( (toMetSp `  N )  e.  * MetSp  ->  (toMetSp `  N )  e.  TopSp )
1210, 11syl 16 . . 3  |-  ( ph  ->  (toMetSp `  N )  e.  TopSp )
13 eqid 2408 . . . 4  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
14 eqid 2408 . . . 4  |-  ( TopOpen `  (toMetSp `  M ) )  =  ( TopOpen `  (toMetSp `  M ) )
15 eqid 2408 . . . 4  |-  ( TopOpen `  (toMetSp `  N ) )  =  ( TopOpen `  (toMetSp `  N ) )
16 eqid 2408 . . . 4  |-  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )
1713, 14, 15, 16xpstopn 17801 . . 3  |-  ( ( (toMetSp `  M )  e.  TopSp  /\  (toMetSp `  N
)  e.  TopSp )  -> 
( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  =  ( ( TopOpen `  (toMetSp `  M ) )  tX  ( TopOpen `  (toMetSp `  N
) ) ) )
186, 12, 17syl2anc 643 . 2  |-  ( ph  ->  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  =  ( ( TopOpen `  (toMetSp `  M ) )  tX  ( TopOpen `  (toMetSp `  N
) ) ) )
19 tmsxpsmopn.l . . 3  |-  L  =  ( MetOpen `  P )
2013xpsxms 18521 . . . . . 6  |-  ( ( (toMetSp `  M )  e.  * MetSp  /\  (toMetSp `  N )  e.  * MetSp )  ->  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  e.  * MetSp )
214, 10, 20syl2anc 643 . . . . 5  |-  ( ph  ->  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
)  e.  * MetSp )
22 eqid 2408 . . . . . 6  |-  ( Base `  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) )  =  (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )
23 tmsxps.p . . . . . . 7  |-  P  =  ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )
2423reseq1i 5105 . . . . . 6  |-  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  =  ( ( dist `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
2516, 22, 24xmstopn 18438 . . . . 5  |-  ( ( (toMetSp `  M )  X.s  (toMetSp `  N ) )  e.  * MetSp  ->  ( TopOpen
`  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
) )  =  (
MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) ) )
2621, 25syl 16 . . . 4  |-  ( ph  ->  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  =  ( MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) ) )
27 eqid 2408 . . . . . . 7  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
28 eqid 2408 . . . . . . 7  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
2913, 27, 28, 4, 10, 23xpsdsfn2 18365 . . . . . 6  |-  ( ph  ->  P  Fn  ( (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
30 fnresdm 5517 . . . . . 6  |-  ( 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 )
3129, 30syl 16 . . . . 5  |-  ( ph  ->  ( P  |`  (
( Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  =  P )
3231fveq2d 5695 . . . 4  |-  ( ph  ->  ( MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) )  =  ( MetOpen `  P )
)
3326, 32eqtr2d 2441 . . 3  |-  ( ph  ->  ( MetOpen `  P )  =  ( TopOpen `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
3419, 33syl5eq 2452 . 2  |-  ( ph  ->  L  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
35 tmsxpsmopn.j . . . . 5  |-  J  =  ( MetOpen `  M )
362, 35tmstopn 18472 . . . 4  |-  ( M  e.  ( * Met `  X )  ->  J  =  ( TopOpen `  (toMetSp `  M ) ) )
371, 36syl 16 . . 3  |-  ( ph  ->  J  =  ( TopOpen `  (toMetSp `  M ) ) )
38 tmsxpsmopn.k . . . . 5  |-  K  =  ( MetOpen `  N )
398, 38tmstopn 18472 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
407, 39syl 16 . . 3  |-  ( ph  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
4137, 40oveq12d 6062 . 2  |-  ( ph  ->  ( J  tX  K
)  =  ( (
TopOpen `  (toMetSp `  M
) )  tX  ( TopOpen
`  (toMetSp `  N )
) ) )
4218, 34, 413eqtr4d 2450 1  |-  ( ph  ->  L  =  ( J 
tX  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    X. cxp 4839    |` cres 4843    Fn wfn 5412   ` cfv 5417  (class class class)co 6044   Basecbs 13428   distcds 13497   TopOpenctopn 13608    X.s cxps 13691   * Metcxmt 16645   MetOpencmopn 16650   TopSpctps 16920    tX ctx 17549   *
MetSpcxme 18304  toMetSpctmt 18306
This theorem is referenced by:  txmetcnp  18534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-map 6983  df-ixp 7027  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-fi 7378  df-sup 7408  df-oi 7439  df-card 7786  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-icc 10883  df-fz 11004  df-fzo 11095  df-seq 11283  df-hash 11578  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-tset 13507  df-ple 13508  df-ds 13510  df-hom 13512  df-cco 13513  df-rest 13609  df-topn 13610  df-topgen 13626  df-pt 13627  df-prds 13630  df-xrs 13685  df-0g 13686  df-gsum 13687  df-qtop 13692  df-imas 13693  df-xps 13695  df-mre 13770  df-mrc 13771  df-acs 13773  df-mnd 14649  df-submnd 14698  df-mulg 14774  df-cntz 15075  df-cmn 15373  df-psmet 16653  df-xmet 16654  df-bl 16656  df-mopn 16657  df-top 16922  df-bases 16924  df-topon 16925  df-topsp 16926  df-cn 17249  df-cnp 17250  df-tx 17551  df-hmeo 17744  df-xms 18307  df-tms 18309
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