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Theorem tmsxpsmopn 18179
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 2358 . . . . . 6  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
32tmsxms 18128 . . . . 5  |-  ( M  e.  ( * Met `  X )  ->  (toMetSp `  M )  e.  * MetSp )
41, 3syl 15 . . . 4  |-  ( ph  ->  (toMetSp `  M )  e.  * MetSp )
5 xmstps 18095 . . . 4  |-  ( (toMetSp `  M )  e.  * MetSp  ->  (toMetSp `  M )  e.  TopSp )
64, 5syl 15 . . 3  |-  ( ph  ->  (toMetSp `  M )  e.  TopSp )
7 tmsxps.2 . . . . 5  |-  ( ph  ->  N  e.  ( * Met `  Y ) )
8 eqid 2358 . . . . . 6  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
98tmsxms 18128 . . . . 5  |-  ( N  e.  ( * Met `  Y )  ->  (toMetSp `  N )  e.  * MetSp )
107, 9syl 15 . . . 4  |-  ( ph  ->  (toMetSp `  N )  e.  * MetSp )
11 xmstps 18095 . . . 4  |-  ( (toMetSp `  N )  e.  * MetSp  ->  (toMetSp `  N )  e.  TopSp )
1210, 11syl 15 . . 3  |-  ( ph  ->  (toMetSp `  N )  e.  TopSp )
13 eqid 2358 . . . 4  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
14 eqid 2358 . . . 4  |-  ( TopOpen `  (toMetSp `  M ) )  =  ( TopOpen `  (toMetSp `  M ) )
15 eqid 2358 . . . 4  |-  ( TopOpen `  (toMetSp `  N ) )  =  ( TopOpen `  (toMetSp `  N ) )
16 eqid 2358 . . . 4  |-  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )
1713, 14, 15, 16xpstopn 17603 . . 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 642 . 2  |-  ( ph  ->  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  =  ( ( TopOpen `  (toMetSp `  M ) )  tX  ( TopOpen `  (toMetSp `  N
) ) ) )
19 tmsxpsmopn.l . . 3  |-  L  =  ( MetOpen `  P )
2013xpsxms 18176 . . . . . 6  |-  ( ( (toMetSp `  M )  e.  * MetSp  /\  (toMetSp `  N )  e.  * MetSp )  ->  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  e.  * MetSp )
214, 10, 20syl2anc 642 . . . . 5  |-  ( ph  ->  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
)  e.  * MetSp )
22 eqid 2358 . . . . . 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 5030 . . . . . 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 18093 . . . . 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 15 . . . 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 2358 . . . . . . 7  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
28 eqid 2358 . . . . . . 7  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
2913, 27, 28, 4, 10, 23xpsdsfn2 18038 . . . . . 6  |-  ( ph  ->  P  Fn  ( (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
30 fnresdm 5432 . . . . . 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 15 . . . . 5  |-  ( ph  ->  ( P  |`  (
( Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )  =  P )
3231fveq2d 5609 . . . 4  |-  ( ph  ->  ( MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) )  =  ( MetOpen `  P )
)
3326, 32eqtr2d 2391 . . 3  |-  ( ph  ->  ( MetOpen `  P )  =  ( TopOpen `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
3419, 33syl5eq 2402 . 2  |-  ( ph  ->  L  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
35 tmsxpsmopn.j . . . . 5  |-  J  =  ( MetOpen `  M )
362, 35tmstopn 18127 . . . 4  |-  ( M  e.  ( * Met `  X )  ->  J  =  ( TopOpen `  (toMetSp `  M ) ) )
371, 36syl 15 . . 3  |-  ( ph  ->  J  =  ( TopOpen `  (toMetSp `  M ) ) )
38 tmsxpsmopn.k . . . . 5  |-  K  =  ( MetOpen `  N )
398, 38tmstopn 18127 . . . 4  |-  ( N  e.  ( * Met `  Y )  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
407, 39syl 15 . . 3  |-  ( ph  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
4137, 40oveq12d 5960 . 2  |-  ( ph  ->  ( J  tX  K
)  =  ( (
TopOpen `  (toMetSp `  M
) )  tX  ( TopOpen
`  (toMetSp `  N )
) ) )
4218, 34, 413eqtr4d 2400 1  |-  ( ph  ->  L  =  ( J 
tX  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    X. cxp 4766    |` cres 4770    Fn wfn 5329   ` cfv 5334  (class class class)co 5942   Basecbs 13239   distcds 13308   TopOpenctopn 13419    X.s cxps 13502   * Metcxmt 16462   MetOpencmopn 16467   TopSpctps 16734    tX ctx 17355   *
MetSpcxme 17978  toMetSpctmt 17980
This theorem is referenced by:  txmetcnp  18189
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-icc 10752  df-fz 10872  df-fzo 10960  df-seq 11136  df-hash 11428  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-hom 13323  df-cco 13324  df-rest 13420  df-topn 13421  df-topgen 13437  df-pt 13438  df-prds 13441  df-xrs 13496  df-0g 13497  df-gsum 13498  df-qtop 13503  df-imas 13504  df-xps 13506  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-submnd 14509  df-mulg 14585  df-cntz 14886  df-cmn 15184  df-xmet 16469  df-bl 16471  df-mopn 16472  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cn 17057  df-cnp 17058  df-tx 17357  df-hmeo 17546  df-xms 17981  df-tms 17983
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