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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 2725 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
3 | 1, 2 | pwsval 17487 | . 2 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
4 | fvexd 6911 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → (Scalar‘𝑅) ∈ V) | |
5 | simpr 483 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝐼 ∈ Fin) | |
6 | fconst6g 6786 | . . . 4 ⊢ (𝑅 ∈ MetSp → (𝐼 × {𝑅}):𝐼⟶MetSp) | |
7 | 6 | adantr 479 | . . 3 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → (𝐼 × {𝑅}):𝐼⟶MetSp) |
8 | eqid 2725 | . . . 4 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
9 | 8 | prdsms 24501 | . . 3 ⊢ (((Scalar‘𝑅) ∈ V ∧ 𝐼 ∈ Fin ∧ (𝐼 × {𝑅}):𝐼⟶MetSp) → ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) ∈ MetSp) |
10 | 4, 5, 7, 9 | syl3anc 1368 | . 2 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) ∈ MetSp) |
11 | 3, 10 | eqeltrd 2825 | 1 ⊢ ((𝑅 ∈ MetSp ∧ 𝐼 ∈ Fin) → 𝑌 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 {csn 4630 × cxp 5676 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 Fincfn 8964 Scalarcsca 17255 Xscprds 17446 ↑s cpws 17447 MetSpcms 24285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9441 df-sup 9472 df-inf 9473 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-icc 13371 df-fz 13525 df-struct 17135 df-slot 17170 df-ndx 17182 df-base 17200 df-plusg 17265 df-mulr 17266 df-sca 17268 df-vsca 17269 df-ip 17270 df-tset 17271 df-ple 17272 df-ds 17274 df-hom 17276 df-cco 17277 df-rest 17423 df-topn 17424 df-topgen 17444 df-pt 17445 df-prds 17448 df-pws 17450 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-top 22857 df-topon 22874 df-topsp 22896 df-bases 22910 df-xms 24287 df-ms 24288 |
This theorem is referenced by: (None) |
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