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Theorem tmsxpsmopn 23123
 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 2820 . . . . . 6 (toMetSp‘𝑀) = (toMetSp‘𝑀)
32tmsxms 23072 . . . . 5 (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp)
41, 3syl 17 . . . 4 (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp)
5 xmstps 23039 . . . 4 ((toMetSp‘𝑀) ∈ ∞MetSp → (toMetSp‘𝑀) ∈ TopSp)
64, 5syl 17 . . 3 (𝜑 → (toMetSp‘𝑀) ∈ TopSp)
7 tmsxps.2 . . . . 5 (𝜑𝑁 ∈ (∞Met‘𝑌))
8 eqid 2820 . . . . . 6 (toMetSp‘𝑁) = (toMetSp‘𝑁)
98tmsxms 23072 . . . . 5 (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp)
107, 9syl 17 . . . 4 (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp)
11 xmstps 23039 . . . 4 ((toMetSp‘𝑁) ∈ ∞MetSp → (toMetSp‘𝑁) ∈ TopSp)
1210, 11syl 17 . . 3 (𝜑 → (toMetSp‘𝑁) ∈ TopSp)
13 eqid 2820 . . . 4 ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁))
14 eqid 2820 . . . 4 (TopOpen‘(toMetSp‘𝑀)) = (TopOpen‘(toMetSp‘𝑀))
15 eqid 2820 . . . 4 (TopOpen‘(toMetSp‘𝑁)) = (TopOpen‘(toMetSp‘𝑁))
16 eqid 2820 . . . 4 (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))
1713, 14, 15, 16xpstopn 22396 . . 3 (((toMetSp‘𝑀) ∈ TopSp ∧ (toMetSp‘𝑁) ∈ TopSp) → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
186, 12, 17syl2anc 586 . 2 (𝜑 → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
19 tmsxpsmopn.l . . 3 𝐿 = (MetOpen‘𝑃)
2013xpsxms 23120 . . . . . 6 (((toMetSp‘𝑀) ∈ ∞MetSp ∧ (toMetSp‘𝑁) ∈ ∞MetSp) → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp)
214, 10, 20syl2anc 586 . . . . 5 (𝜑 → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp)
22 eqid 2820 . . . . . 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 5825 . . . . . 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 23037 . . . . 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 2820 . . . . . . 7 (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀))
28 eqid 2820 . . . . . . 7 (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁))
2913, 27, 28, 4, 10, 23xpsdsfn2 22964 . . . . . 6 (𝜑𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))
30 fnresdm 6442 . . . . . 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 6650 . . . 4 (𝜑 → (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))) = (MetOpen‘𝑃))
3326, 32eqtr2d 2856 . . 3 (𝜑 → (MetOpen‘𝑃) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))
3419, 33syl5eq 2867 . 2 (𝜑𝐿 = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))
35 tmsxpsmopn.j . . . . 5 𝐽 = (MetOpen‘𝑀)
362, 35tmstopn 23071 . . . 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 23071 . . . 4 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (TopOpen‘(toMetSp‘𝑁)))
407, 39syl 17 . . 3 (𝜑𝐾 = (TopOpen‘(toMetSp‘𝑁)))
4137, 40oveq12d 7151 . 2 (𝜑 → (𝐽 ×t 𝐾) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
4218, 34, 413eqtr4d 2865 1 (𝜑𝐿 = (𝐽 ×t 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   × cxp 5529   ↾ cres 5533   Fn wfn 6326  ‘cfv 6331  (class class class)co 7133  Basecbs 16462  distcds 16553  TopOpenctopn 16674   ×s cxps 16758  ∞Metcxmet 20506  MetOpencmopn 20511  TopSpctps 21516   ×t ctx 22144  ∞MetSpcxms 22903  toMetSpctms 22905 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-om 7559  df-1st 7667  df-2nd 7668  df-supp 7809  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-map 8386  df-ixp 8440  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-fsupp 8812  df-fi 8853  df-sup 8884  df-inf 8885  df-oi 8952  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-z 11961  df-dec 12078  df-uz 12223  df-q 12328  df-rp 12369  df-xneg 12486  df-xadd 12487  df-xmul 12488  df-icc 12724  df-fz 12877  df-fzo 13018  df-seq 13354  df-hash 13676  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-mulr 16558  df-sca 16560  df-vsca 16561  df-ip 16562  df-tset 16563  df-ple 16564  df-ds 16566  df-hom 16568  df-cco 16569  df-rest 16675  df-topn 16676  df-0g 16694  df-gsum 16695  df-topgen 16696  df-pt 16697  df-prds 16700  df-xrs 16754  df-qtop 16759  df-imas 16760  df-xps 16762  df-mre 16836  df-mrc 16837  df-acs 16839  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-submnd 17936  df-mulg 18204  df-cntz 18426  df-cmn 18887  df-psmet 20513  df-xmet 20514  df-bl 20516  df-mopn 20517  df-top 21478  df-topon 21495  df-topsp 21517  df-bases 21530  df-cn 21811  df-cnp 21812  df-tx 22146  df-hmeo 22339  df-xms 22906  df-tms 22908 This theorem is referenced by:  txmetcnp  23133
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