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Theorem tmsxpsmopn 20913
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 2443 . . . . . 6  |-  (toMetSp `  M
)  =  (toMetSp `  M
)
32tmsxms 20862 . . . . 5  |-  ( M  e.  ( *Met `  X )  ->  (toMetSp `  M )  e.  *MetSp )
41, 3syl 16 . . . 4  |-  ( ph  ->  (toMetSp `  M )  e.  *MetSp )
5 xmstps 20829 . . . 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 2443 . . . . . 6  |-  (toMetSp `  N
)  =  (toMetSp `  N
)
98tmsxms 20862 . . . . 5  |-  ( N  e.  ( *Met `  Y )  ->  (toMetSp `  N )  e.  *MetSp )
107, 9syl 16 . . . 4  |-  ( ph  ->  (toMetSp `  N )  e.  *MetSp )
11 xmstps 20829 . . . 4  |-  ( (toMetSp `  N )  e.  *MetSp  ->  (toMetSp `  N )  e.  TopSp )
1210, 11syl 16 . . 3  |-  ( ph  ->  (toMetSp `  N )  e.  TopSp )
13 eqid 2443 . . . 4  |-  ( (toMetSp `  M )  X.s  (toMetSp `  N
) )  =  ( (toMetSp `  M )  X.s  (toMetSp `  N ) )
14 eqid 2443 . . . 4  |-  ( TopOpen `  (toMetSp `  M ) )  =  ( TopOpen `  (toMetSp `  M ) )
15 eqid 2443 . . . 4  |-  ( TopOpen `  (toMetSp `  N ) )  =  ( TopOpen `  (toMetSp `  N ) )
16 eqid 2443 . . . 4  |-  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) )
1713, 14, 15, 16xpstopn 20186 . . 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 20910 . . . . . 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 2443 . . . . . 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 5259 . . . . . 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 20827 . . . . 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 2443 . . . . . . 7  |-  ( Base `  (toMetSp `  M )
)  =  ( Base `  (toMetSp `  M )
)
28 eqid 2443 . . . . . . 7  |-  ( Base `  (toMetSp `  N )
)  =  ( Base `  (toMetSp `  N )
)
2913, 27, 28, 4, 10, 23xpsdsfn2 20754 . . . . . 6  |-  ( ph  ->  P  Fn  ( (
Base `  ( (toMetSp `  M )  X.s  (toMetSp `  N
) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) )
30 fnresdm 5680 . . . . . 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 5860 . . . 4  |-  ( ph  ->  ( MetOpen `  ( P  |`  ( ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) )  X.  ( Base `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) ) ) )  =  ( MetOpen `  P )
)
3326, 32eqtr2d 2485 . . 3  |-  ( ph  ->  ( MetOpen `  P )  =  ( TopOpen `  (
(toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
3419, 33syl5eq 2496 . 2  |-  ( ph  ->  L  =  ( TopOpen `  ( (toMetSp `  M )  X.s  (toMetSp `  N ) ) ) )
35 tmsxpsmopn.j . . . . 5  |-  J  =  ( MetOpen `  M )
362, 35tmstopn 20861 . . . 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 20861 . . . 4  |-  ( N  e.  ( *Met `  Y )  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
407, 39syl 16 . . 3  |-  ( ph  ->  K  =  ( TopOpen `  (toMetSp `  N ) ) )
4137, 40oveq12d 6299 . 2  |-  ( ph  ->  ( J  tX  K
)  =  ( (
TopOpen `  (toMetSp `  M
) )  tX  ( TopOpen
`  (toMetSp `  N )
) ) )
4218, 34, 413eqtr4d 2494 1  |-  ( ph  ->  L  =  ( J 
tX  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804    X. cxp 4987    |` cres 4991    Fn wfn 5573   ` cfv 5578  (class class class)co 6281   Basecbs 14509   distcds 14583   TopOpenctopn 14696    X.s cxps 14780   *Metcxmt 18277   MetOpencmopn 18282   TopSpctps 19270    tX ctx 19934   *MetSpcxme 20693  toMetSpctmt 20695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-icc 11545  df-fz 11682  df-fzo 11804  df-seq 12087  df-hash 12385  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-bl 18288  df-mopn 18289  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cn 19601  df-cnp 19602  df-tx 19936  df-hmeo 20129  df-xms 20696  df-tms 20698
This theorem is referenced by:  txmetcnp  20923
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