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Mirrors > Home > MPE Home > Th. List > rrxmetfi | Structured version Visualization version GIF version |
Description: Euclidean space is a metric space. Finite dimensional version. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
rrxmetfi.1 | ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) |
Ref | Expression |
---|---|
rrxmetfi | ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} | |
2 | rrxmetfi.1 | . . 3 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼)) | |
3 | 1, 2 | rrxmet 24162 | . 2 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘{ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0})) |
4 | eqid 2738 | . . . . 5 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼) | |
5 | eqid 2738 | . . . . 5 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼)) | |
6 | 4, 5 | rrxbase 24142 | . . . 4 ⊢ (𝐼 ∈ Fin → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}) |
7 | id 22 | . . . . 5 ⊢ (𝐼 ∈ Fin → 𝐼 ∈ Fin) | |
8 | 7, 4, 5 | rrxbasefi 24164 | . . . 4 ⊢ (𝐼 ∈ Fin → (Base‘(ℝ^‘𝐼)) = (ℝ ↑m 𝐼)) |
9 | 6, 8 | eqtr3d 2775 | . . 3 ⊢ (𝐼 ∈ Fin → {ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0} = (ℝ ↑m 𝐼)) |
10 | 9 | fveq2d 6680 | . 2 ⊢ (𝐼 ∈ Fin → (Met‘{ℎ ∈ (ℝ ↑m 𝐼) ∣ ℎ finSupp 0}) = (Met‘(ℝ ↑m 𝐼))) |
11 | 3, 10 | eleqtrd 2835 | 1 ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘(ℝ ↑m 𝐼))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3057 class class class wbr 5030 ‘cfv 6339 (class class class)co 7172 ↑m cmap 8439 Fincfn 8557 finSupp cfsupp 8908 ℝcr 10616 0cc0 10617 Basecbs 16588 distcds 16679 Metcmet 20205 ℝ^crrx 24137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-inf2 9179 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 ax-addf 10696 ax-mulf 10697 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-supp 7859 df-tpos 7923 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-map 8441 df-ixp 8510 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-fsupp 8909 df-sup 8981 df-oi 9049 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-rp 12475 df-ico 12829 df-fz 12984 df-fzo 13127 df-seq 13463 df-exp 13524 df-hash 13785 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-clim 14937 df-sum 15138 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-starv 16685 df-sca 16686 df-vsca 16687 df-ip 16688 df-tset 16689 df-ple 16690 df-ds 16692 df-unif 16693 df-hom 16694 df-cco 16695 df-0g 16820 df-gsum 16821 df-prds 16826 df-pws 16828 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-mhm 18074 df-grp 18224 df-minusg 18225 df-sbg 18226 df-subg 18396 df-ghm 18476 df-cntz 18567 df-cmn 19028 df-abl 19029 df-mgp 19361 df-ur 19373 df-ring 19420 df-cring 19421 df-oppr 19497 df-dvdsr 19515 df-unit 19516 df-invr 19546 df-dvr 19557 df-rnghom 19591 df-drng 19625 df-field 19626 df-subrg 19654 df-staf 19737 df-srng 19738 df-lmod 19757 df-lss 19825 df-sra 20065 df-rgmod 20066 df-met 20213 df-cnfld 20220 df-refld 20423 df-dsmm 20550 df-frlm 20565 df-nm 23337 df-tng 23339 df-tcph 23923 df-rrx 24139 |
This theorem is referenced by: qndenserrnbllem 43399 qndenserrnbl 43400 qndenserrnopnlem 43402 rrndsmet 43407 hoiqssbllem2 43725 hoiqssbl 43727 opnvonmbllem2 43735 rrxsphere 45657 |
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