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Mirrors > Home > MPE Home > Th. List > pwsms | Structured version Visualization version GIF version |
Description: A power of a metric space 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 2738 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
3 | 1, 2 | pwsval 17195 | . 2 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
4 | fvexd 6791 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → (Scalar‘𝑅) ∈ V) | |
5 | simpr 485 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin) | |
6 | fconst6g 6665 | . . . 4 ⊢ (𝑅 ∈ MetSp → (𝐼 × {𝑅}):𝐼⟶MetSp) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → (𝐼 × {𝑅}):𝐼⟶MetSp) |
8 | eqid 2738 | . . . 4 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
9 | 8 | prdsms 23685 | . . 3 ⊢ (((Scalar‘𝑅) ∈ V ∧ 𝐼 ∈ Fin ∧ (𝐼 × {𝑅}):𝐼⟶MetSp) → ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) ∈ MetSp) |
10 | 4, 5, 7, 9 | syl3anc 1370 | . 2 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) ∈ MetSp) |
11 | 3, 10 | eqeltrd 2839 | 1 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3431 {csn 4563 × cxp 5589 ⟶wf 6431 ‘cfv 6435 (class class class)co 7277 Fincfn 8731 Scalarcsca 16963 Xscprds 17154 ↑s cpws 17155 MetSpcms 23469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-er 8496 df-map 8615 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fi 9168 df-sup 9199 df-inf 9200 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 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 12436 df-uz 12581 df-q 12687 df-rp 12729 df-xneg 12846 df-xadd 12847 df-xmul 12848 df-icc 13084 df-fz 13238 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-topgen 17152 df-pt 17153 df-prds 17156 df-pws 17158 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-xms 23471 df-ms 23472 |
This theorem is referenced by: (None) |
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