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Theorem tmsxpsmopn 22152
 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 2610 . . . . . 6 (toMetSp‘𝑀) = (toMetSp‘𝑀)
32tmsxms 22101 . . . . 5 (𝑀 ∈ (∞Met‘𝑋) → (toMetSp‘𝑀) ∈ ∞MetSp)
41, 3syl 17 . . . 4 (𝜑 → (toMetSp‘𝑀) ∈ ∞MetSp)
5 xmstps 22068 . . . 4 ((toMetSp‘𝑀) ∈ ∞MetSp → (toMetSp‘𝑀) ∈ TopSp)
64, 5syl 17 . . 3 (𝜑 → (toMetSp‘𝑀) ∈ TopSp)
7 tmsxps.2 . . . . 5 (𝜑𝑁 ∈ (∞Met‘𝑌))
8 eqid 2610 . . . . . 6 (toMetSp‘𝑁) = (toMetSp‘𝑁)
98tmsxms 22101 . . . . 5 (𝑁 ∈ (∞Met‘𝑌) → (toMetSp‘𝑁) ∈ ∞MetSp)
107, 9syl 17 . . . 4 (𝜑 → (toMetSp‘𝑁) ∈ ∞MetSp)
11 xmstps 22068 . . . 4 ((toMetSp‘𝑁) ∈ ∞MetSp → (toMetSp‘𝑁) ∈ TopSp)
1210, 11syl 17 . . 3 (𝜑 → (toMetSp‘𝑁) ∈ TopSp)
13 eqid 2610 . . . 4 ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) = ((toMetSp‘𝑀) ×s (toMetSp‘𝑁))
14 eqid 2610 . . . 4 (TopOpen‘(toMetSp‘𝑀)) = (TopOpen‘(toMetSp‘𝑀))
15 eqid 2610 . . . 4 (TopOpen‘(toMetSp‘𝑁)) = (TopOpen‘(toMetSp‘𝑁))
16 eqid 2610 . . . 4 (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))
1713, 14, 15, 16xpstopn 21425 . . 3 (((toMetSp‘𝑀) ∈ TopSp ∧ (toMetSp‘𝑁) ∈ TopSp) → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
186, 12, 17syl2anc 691 . 2 (𝜑 → (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
19 tmsxpsmopn.l . . 3 𝐿 = (MetOpen‘𝑃)
2013xpsxms 22149 . . . . . 6 (((toMetSp‘𝑀) ∈ ∞MetSp ∧ (toMetSp‘𝑁) ∈ ∞MetSp) → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp)
214, 10, 20syl2anc 691 . . . . 5 (𝜑 → ((toMetSp‘𝑀) ×s (toMetSp‘𝑁)) ∈ ∞MetSp)
22 eqid 2610 . . . . . 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 5313 . . . . . 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 22066 . . . . 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 2610 . . . . . . 7 (Base‘(toMetSp‘𝑀)) = (Base‘(toMetSp‘𝑀))
28 eqid 2610 . . . . . . 7 (Base‘(toMetSp‘𝑁)) = (Base‘(toMetSp‘𝑁))
2913, 27, 28, 4, 10, 23xpsdsfn2 21993 . . . . . 6 (𝜑𝑃 Fn ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))
30 fnresdm 5914 . . . . . 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 6107 . . . 4 (𝜑 → (MetOpen‘(𝑃 ↾ ((Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) × (Base‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁)))))) = (MetOpen‘𝑃))
3326, 32eqtr2d 2645 . . 3 (𝜑 → (MetOpen‘𝑃) = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))
3419, 33syl5eq 2656 . 2 (𝜑𝐿 = (TopOpen‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))))
35 tmsxpsmopn.j . . . . 5 𝐽 = (MetOpen‘𝑀)
362, 35tmstopn 22100 . . . 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 22100 . . . 4 (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (TopOpen‘(toMetSp‘𝑁)))
407, 39syl 17 . . 3 (𝜑𝐾 = (TopOpen‘(toMetSp‘𝑁)))
4137, 40oveq12d 6567 . 2 (𝜑 → (𝐽 ×t 𝐾) = ((TopOpen‘(toMetSp‘𝑀)) ×t (TopOpen‘(toMetSp‘𝑁))))
4218, 34, 413eqtr4d 2654 1 (𝜑𝐿 = (𝐽 ×t 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   × cxp 5036   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TopOpenctopn 15905   ×s cxps 15989  ∞Metcxmt 19552  MetOpencmopn 19557  TopSpctps 20519   ×t ctx 21173  ∞MetSpcxme 21932  toMetSpctmt 21934 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-tx 21175  df-hmeo 21368  df-xms 21935  df-tms 21937 This theorem is referenced by:  txmetcnp  22162
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