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Mirrors > Home > MPE Home > Th. List > pwsms | Structured version Visualization version GIF version |
Description: The product of a finite family of metric spaces is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
pwsms.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
Ref | Expression |
---|---|
pwsms | ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsms.y | . . 3 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
2 | eqid 2778 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
3 | 1, 2 | pwsval 16543 | . 2 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
4 | fvexd 6463 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → (Scalar‘𝑅) ∈ V) | |
5 | simpr 479 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin) | |
6 | fconst6g 6346 | . . . 4 ⊢ (𝑅 ∈ MetSp → (𝐼 × {𝑅}):𝐼⟶MetSp) | |
7 | 6 | adantr 474 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → (𝐼 × {𝑅}):𝐼⟶MetSp) |
8 | eqid 2778 | . . . 4 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
9 | 8 | prdsms 22755 | . . 3 ⊢ (((Scalar‘𝑅) ∈ V ∧ 𝐼 ∈ Fin ∧ (𝐼 × {𝑅}):𝐼⟶MetSp) → ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) ∈ MetSp) |
10 | 4, 5, 7, 9 | syl3anc 1439 | . 2 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) ∈ MetSp) |
11 | 3, 10 | eqeltrd 2859 | 1 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 {csn 4398 × cxp 5355 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 Fincfn 8243 Scalarcsca 16352 Xscprds 16503 ↑s cpws 16504 MetSpcms 22542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fi 8607 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-icc 12499 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-topgen 16501 df-pt 16502 df-prds 16505 df-pws 16507 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-xms 22544 df-ms 22545 |
This theorem is referenced by: (None) |
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