MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tmsxpsmopn Structured version   Visualization version   GIF version

Theorem tmsxpsmopn 23689
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) ×s (toMetSp‘𝑁)))
tmsxps.1 (𝜑𝑀 ∈ (∞Met‘𝑋))
tmsxps.2 (𝜑𝑁 ∈ (∞Met‘𝑌))
tmsxpsmopn.j 𝐽 = (MetOpen‘𝑀)
tmsxpsmopn.k 𝐾 = (MetOpen‘𝑁)
tmsxpsmopn.l 𝐿 = (MetOpen‘𝑃)
Assertion
Ref Expression
tmsxpsmopn (𝜑𝐿 = (𝐽 ×t 𝐾))

Proof of Theorem tmsxpsmopn
StepHypRef Expression
1 tmsxps.1 . . . . 5 (𝜑𝑀 ∈ (∞Met‘𝑋))
2 eqid 2740 . . . . . 6 (toMetSp‘𝑀) = (toMetSp‘𝑀)
32tmsxms 23638 . . . . 5 (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp)
41, 3syl 17 . . . 4 (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp)
5 xmstps 23602 . . . 4 ((toMetSp‘𝑀) ∈ ∞MetSp → (toMetSp‘𝑀) ∈ TopSp)
64, 5syl 17 . . 3 (𝜑 → (toMetSp‘𝑀) ∈ TopSp)
7 tmsxps.2 . . . . 5 (𝜑𝑁 ∈ (∞Met‘𝑌))
8 eqid 2740 . . . . . 6 (toMetSp‘𝑁) = (toMetSp‘𝑁)
98tmsxms 23638 . . . . 5 (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp)
107, 9syl 17 . . . 4 (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp)
11 xmstps 23602 . . . 4 ((toMetSp‘𝑁) ∈ ∞MetSp → (toMetSp‘𝑁) ∈ TopSp)
1210, 11syl 17 . . 3 (𝜑 → (toMetSp‘𝑁) ∈ TopSp)
13 eqid 2740 . . . 4 ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁))
14 eqid 2740 . . . 4 (TopOpen‘(toMetSp‘𝑀)) = (TopOpen‘(toMetSp‘𝑀))
15 eqid 2740 . . . 4 (TopOpen‘(toMetSp‘𝑁)) = (TopOpen‘(toMetSp‘𝑁))
16 eqid 2740 . . . 4 (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))
1713, 14, 15, 16xpstopn 22959 . . 3 (((toMetSp‘𝑀) ∈ TopSp ∧ (toMetSp‘𝑁) ∈ TopSp) → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
186, 12, 17syl2anc 584 . 2 (𝜑 → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
19 tmsxpsmopn.l . . 3 𝐿 = (MetOpen‘𝑃)
2013xpsxms 23686 . . . . . 6 (((toMetSp‘𝑀) ∈ ∞MetSp ∧ (toMetSp‘𝑁) ∈ ∞MetSp) → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp)
214, 10, 20syl2anc 584 . . . . 5 (𝜑 → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp)
22 eqid 2740 . . . . . 6 (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))
23 tmsxps.p . . . . . . 7 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))
2423reseq1i 5885 . . . . . 6 (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) = ((dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))
2516, 22, 24xmstopn 23600 . . . . 5 (((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))))
2621, 25syl 17 . . . 4 (𝜑 → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))))
27 eqid 2740 . . . . . . 7 (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀))
28 eqid 2740 . . . . . . 7 (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁))
2913, 27, 28, 4, 10, 23xpsdsfn2 23527 . . . . . 6 (𝜑𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))
30 fnresdm 6548 . . . . . 6 (𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))) → (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) = 𝑃)
3129, 30syl 17 . . . . 5 (𝜑 → (𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))) = 𝑃)
3231fveq2d 6773 . . . 4 (𝜑 → (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))) = (MetOpen‘𝑃))
3326, 32eqtr2d 2781 . . 3 (𝜑 → (MetOpen‘𝑃) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))
3419, 33eqtrid 2792 . 2 (𝜑𝐿 = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))
35 tmsxpsmopn.j . . . . 5 𝐽 = (MetOpen‘𝑀)
362, 35tmstopn 23637 . . . 4 (𝑀 ∈ (∞Met‘𝑋) → 𝐽 = (TopOpen‘(toMetSp‘𝑀)))
371, 36syl 17 . . 3 (𝜑𝐽 = (TopOpen‘(toMetSp‘𝑀)))
38 tmsxpsmopn.k . . . . 5 𝐾 = (MetOpen‘𝑁)
398, 38tmstopn 23637 . . . 4 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (TopOpen‘(toMetSp‘𝑁)))
407, 39syl 17 . . 3 (𝜑𝐾 = (TopOpen‘(toMetSp‘𝑁)))
4137, 40oveq12d 7287 . 2 (𝜑 → (𝐽 ×t 𝐾) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
4218, 34, 413eqtr4d 2790 1 (𝜑𝐿 = (𝐽 ×t 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110   × cxp 5587  cres 5591   Fn wfn 6426  cfv 6431  (class class class)co 7269  Basecbs 16908  distcds 16967  TopOpenctopn 17128   ×s cxps 17213  ∞Metcxmet 20578  MetOpencmopn 20583  TopSpctps 22077   ×t ctx 22707  ∞MetSpcxms 23466  toMetSpctms 23468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-cnex 10926  ax-resscn 10927  ax-1cn 10928  ax-icn 10929  ax-addcl 10930  ax-addrcl 10931  ax-mulcl 10932  ax-mulrcl 10933  ax-mulcom 10934  ax-addass 10935  ax-mulass 10936  ax-distr 10937  ax-i2m1 10938  ax-1ne0 10939  ax-1rid 10940  ax-rnegex 10941  ax-rrecex 10942  ax-cnre 10943  ax-pre-lttri 10944  ax-pre-lttrn 10945  ax-pre-ltadd 10946  ax-pre-mulgt0 10947  ax-pre-sup 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-isom 6440  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-of 7525  df-om 7705  df-1st 7822  df-2nd 7823  df-supp 7967  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-1o 8286  df-2o 8287  df-er 8479  df-map 8598  df-ixp 8667  df-en 8715  df-dom 8716  df-sdom 8717  df-fin 8718  df-fsupp 9105  df-fi 9146  df-sup 9177  df-inf 9178  df-oi 9245  df-card 9696  df-pnf 11010  df-mnf 11011  df-xr 11012  df-ltxr 11013  df-le 11014  df-sub 11205  df-neg 11206  df-div 11631  df-nn 11972  df-2 12034  df-3 12035  df-4 12036  df-5 12037  df-6 12038  df-7 12039  df-8 12040  df-9 12041  df-n0 12232  df-z 12318  df-dec 12435  df-uz 12580  df-q 12686  df-rp 12728  df-xneg 12845  df-xadd 12846  df-xmul 12847  df-icc 13083  df-fz 13237  df-fzo 13380  df-seq 13718  df-hash 14041  df-struct 16844  df-sets 16861  df-slot 16879  df-ndx 16891  df-base 16909  df-ress 16938  df-plusg 16971  df-mulr 16972  df-sca 16974  df-vsca 16975  df-ip 16976  df-tset 16977  df-ple 16978  df-ds 16980  df-hom 16982  df-cco 16983  df-rest 17129  df-topn 17130  df-0g 17148  df-gsum 17149  df-topgen 17150  df-pt 17151  df-prds 17154  df-xrs 17209  df-qtop 17214  df-imas 17215  df-xps 17217  df-mre 17291  df-mrc 17292  df-acs 17294  df-mgm 18322  df-sgrp 18371  df-mnd 18382  df-submnd 18427  df-mulg 18697  df-cntz 18919  df-cmn 19384  df-psmet 20585  df-xmet 20586  df-bl 20588  df-mopn 20589  df-top 22039  df-topon 22056  df-topsp 22078  df-bases 22092  df-cn 22374  df-cnp 22375  df-tx 22709  df-hmeo 22902  df-xms 23469  df-tms 23471
This theorem is referenced by:  txmetcnp  23699
  Copyright terms: Public domain W3C validator