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Theorem tmsxpsmopn 20254
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 2454 . . . . . 6  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
32tmsxms 20203 . . . . 5  |-  ( M  e.  ( *Met `  X )  ->  (toMetSp `  M )  e.  *MetSp )
41, 3syl 16 . . . 4  |-  ( ph  ->  (toMetSp `  M )  e.  *MetSp )
5 xmstps 20170 . . . 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 2454 . . . . . 6  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
98tmsxms 20203 . . . . 5  |-  ( N  e.  ( *Met `  Y )  ->  (toMetSp `  N )  e.  *MetSp )
107, 9syl 16 . . . 4  |-  ( ph  ->  (toMetSp `  N )  e.  *MetSp )
11 xmstps 20170 . . . 4  |-  ( (toMetSp `  N )  e.  *MetSp  ->  (toMetSp `  N )  e.  TopSp )
1210, 11syl 16 . . 3  |-  ( ph  ->  (toMetSp `  N )  e.  TopSp )
13 eqid 2454 . . . 4  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
14 eqid 2454 . . . 4  |-  ( TopOpen `  (toMetSp `  M ) )  =  ( TopOpen `  (toMetSp `  M ) )
15 eqid 2454 . . . 4  |-  ( TopOpen `  (toMetSp `  N ) )  =  ( TopOpen `  (toMetSp `  N ) )
16 eqid 2454 . . . 4  |-  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )
1713, 14, 15, 16xpstopn 19527 . . 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 661 . 2  |-  ( ph  ->  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  =  ( ( TopOpen `  (toMetSp `  M ) )  tX  ( TopOpen `  (toMetSp `  N
) ) ) )
19 tmsxpsmopn.l . . 3  |-  L  =  ( MetOpen `  P )
2013xpsxms 20251 . . . . . 6  |-  ( ( (toMetSp `  M )  e.  *MetSp  /\  (toMetSp `  N )  e.  *MetSp )  ->  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  e.  *MetSp )
214, 10, 20syl2anc 661 . . . . 5  |-  ( ph  ->  ( (toMetSp `  M
)  X.s  (toMetSp `  N )
)  e.  *MetSp )
22 eqid 2454 . . . . . 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 5217 . . . . . 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 20168 . . . . 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 2454 . . . . . . 7  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
28 eqid 2454 . . . . . . 7  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
2913, 27, 28, 4, 10, 23xpsdsfn2 20095 . . . . . 6  |-  ( ph  ->  P  Fn  ( (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
30 fnresdm 5631 . . . . . 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 5806 . . . 4  |-  ( ph  ->  ( MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) )  =  ( MetOpen `  P )
)
3326, 32eqtr2d 2496 . . 3  |-  ( ph  ->  ( MetOpen `  P )  =  ( TopOpen `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
3419, 33syl5eq 2507 . 2  |-  ( ph  ->  L  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
35 tmsxpsmopn.j . . . . 5  |-  J  =  ( MetOpen `  M )
362, 35tmstopn 20202 . . . 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 20202 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
407, 39syl 16 . . 3  |-  ( ph  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
4137, 40oveq12d 6221 . 2  |-  ( ph  ->  ( J  tX  K
)  =  ( (
TopOpen `  (toMetSp `  M
) )  tX  ( TopOpen
`  (toMetSp `  N )
) ) )
4218, 34, 413eqtr4d 2505 1  |-  ( ph  ->  L  =  ( J 
tX  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    X. cxp 4949    |` cres 4953    Fn wfn 5524   ` cfv 5529  (class class class)co 6203   Basecbs 14296   distcds 14370   TopOpenctopn 14483    X.s cxps 14567   *Metcxmt 17936   MetOpencmopn 17941   TopSpctps 18643    tX ctx 19275   *MetSpcxme 20034  toMetSpctmt 20036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7776  df-sup 7806  df-oi 7839  df-card 8224  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-q 11069  df-rp 11107  df-xneg 11204  df-xadd 11205  df-xmul 11206  df-icc 11422  df-fz 11559  df-fzo 11670  df-seq 11928  df-hash 12225  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-ip 14379  df-tset 14380  df-ple 14381  df-ds 14383  df-hom 14385  df-cco 14386  df-rest 14484  df-topn 14485  df-0g 14503  df-gsum 14504  df-topgen 14505  df-pt 14506  df-prds 14509  df-xrs 14563  df-qtop 14568  df-imas 14569  df-xps 14571  df-mre 14647  df-mrc 14648  df-acs 14650  df-mnd 15538  df-submnd 15588  df-mulg 15671  df-cntz 15958  df-cmn 16404  df-psmet 17944  df-xmet 17945  df-bl 17947  df-mopn 17948  df-top 18645  df-bases 18647  df-topon 18648  df-topsp 18649  df-cn 18973  df-cnp 18974  df-tx 19277  df-hmeo 19470  df-xms 20037  df-tms 20039
This theorem is referenced by:  txmetcnp  20264
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