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Theorem tmsxpsmopn 23076
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 2821 . . . . . 6 (toMetSp‘𝑀) = (toMetSp‘𝑀)
32tmsxms 23025 . . . . 5 (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp)
41, 3syl 17 . . . 4 (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp)
5 xmstps 22992 . . . 4 ((toMetSp‘𝑀) ∈ ∞MetSp → (toMetSp‘𝑀) ∈ TopSp)
64, 5syl 17 . . 3 (𝜑 → (toMetSp‘𝑀) ∈ TopSp)
7 tmsxps.2 . . . . 5 (𝜑𝑁 ∈ (∞Met‘𝑌))
8 eqid 2821 . . . . . 6 (toMetSp‘𝑁) = (toMetSp‘𝑁)
98tmsxms 23025 . . . . 5 (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp)
107, 9syl 17 . . . 4 (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp)
11 xmstps 22992 . . . 4 ((toMetSp‘𝑁) ∈ ∞MetSp → (toMetSp‘𝑁) ∈ TopSp)
1210, 11syl 17 . . 3 (𝜑 → (toMetSp‘𝑁) ∈ TopSp)
13 eqid 2821 . . . 4 ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁))
14 eqid 2821 . . . 4 (TopOpen‘(toMetSp‘𝑀)) = (TopOpen‘(toMetSp‘𝑀))
15 eqid 2821 . . . 4 (TopOpen‘(toMetSp‘𝑁)) = (TopOpen‘(toMetSp‘𝑁))
16 eqid 2821 . . . 4 (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))
1713, 14, 15, 16xpstopn 22350 . . 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 23073 . . . . . 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 2821 . . . . . 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 5843 . . . . . 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 22990 . . . . 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 2821 . . . . . . 7 (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀))
28 eqid 2821 . . . . . . 7 (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁))
2913, 27, 28, 4, 10, 23xpsdsfn2 22917 . . . . . 6 (𝜑𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))
30 fnresdm 6460 . . . . . 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 6668 . . . 4 (𝜑 → (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))) = (MetOpen‘𝑃))
3326, 32eqtr2d 2857 . . 3 (𝜑 → (MetOpen‘𝑃) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))
3419, 33syl5eq 2868 . 2 (𝜑𝐿 = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))
35 tmsxpsmopn.j . . . . 5 𝐽 = (MetOpen‘𝑀)
362, 35tmstopn 23024 . . . 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 23024 . . . 4 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (TopOpen‘(toMetSp‘𝑁)))
407, 39syl 17 . . 3 (𝜑𝐾 = (TopOpen‘(toMetSp‘𝑁)))
4137, 40oveq12d 7163 . 2 (𝜑 → (𝐽 ×t 𝐾) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
4218, 34, 413eqtr4d 2866 1 (𝜑𝐿 = (𝐽 ×t 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105   × cxp 5547  cres 5551   Fn wfn 6344  cfv 6349  (class class class)co 7145  Basecbs 16473  distcds 16564  TopOpenctopn 16685   ×s cxps 16769  ∞Metcxmet 20460  MetOpencmopn 20465  TopSpctps 21470   ×t ctx 22098  ∞MetSpcxms 22856  toMetSpctms 22858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-map 8398  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fsupp 8823  df-fi 8864  df-sup 8895  df-inf 8896  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-q 12338  df-rp 12380  df-xneg 12497  df-xadd 12498  df-xmul 12499  df-icc 12735  df-fz 12883  df-fzo 13024  df-seq 13360  df-hash 13681  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-mulr 16569  df-sca 16571  df-vsca 16572  df-ip 16573  df-tset 16574  df-ple 16575  df-ds 16577  df-hom 16579  df-cco 16580  df-rest 16686  df-topn 16687  df-0g 16705  df-gsum 16706  df-topgen 16707  df-pt 16708  df-prds 16711  df-xrs 16765  df-qtop 16770  df-imas 16771  df-xps 16773  df-mre 16847  df-mrc 16848  df-acs 16850  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-submnd 17947  df-mulg 18165  df-cntz 18387  df-cmn 18839  df-psmet 20467  df-xmet 20468  df-bl 20470  df-mopn 20471  df-top 21432  df-topon 21449  df-topsp 21471  df-bases 21484  df-cn 21765  df-cnp 21766  df-tx 22100  df-hmeo 22293  df-xms 22859  df-tms 22861
This theorem is referenced by:  txmetcnp  23086
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