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Mirrors > Home > MPE Home > Th. List > ehlbase | Structured version Visualization version GIF version |
Description: The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
Ref | Expression |
---|---|
ehlval.e | ⊢ 𝐸 = (𝔼hil‘𝑁) |
Ref | Expression |
---|---|
ehlbase | ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...𝑁)) = (Base‘𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ehlval.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘𝑁) | |
2 | 1 | ehlval 23239 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) |
3 | 2 | fveq2d 6233 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Base‘𝐸) = (Base‘(ℝ^‘(1...𝑁)))) |
4 | rabid2 3148 | . . . 4 ⊢ ((ℝ ↑𝑚 (1...𝑁)) = {𝑓 ∈ (ℝ ↑𝑚 (1...𝑁)) ∣ 𝑓 finSupp 0} ↔ ∀𝑓 ∈ (ℝ ↑𝑚 (1...𝑁))𝑓 finSupp 0) | |
5 | elmapi 7921 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑𝑚 (1...𝑁)) → 𝑓:(1...𝑁)⟶ℝ) | |
6 | fzfid 12812 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑𝑚 (1...𝑁)) → (1...𝑁) ∈ Fin) | |
7 | 0red 10079 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑𝑚 (1...𝑁)) → 0 ∈ ℝ) | |
8 | 5, 6, 7 | fdmfifsupp 8326 | . . . 4 ⊢ (𝑓 ∈ (ℝ ↑𝑚 (1...𝑁)) → 𝑓 finSupp 0) |
9 | 4, 8 | mprgbir 2956 | . . 3 ⊢ (ℝ ↑𝑚 (1...𝑁)) = {𝑓 ∈ (ℝ ↑𝑚 (1...𝑁)) ∣ 𝑓 finSupp 0} |
10 | ovex 6718 | . . . 4 ⊢ (1...𝑁) ∈ V | |
11 | eqid 2651 | . . . . 5 ⊢ (ℝ^‘(1...𝑁)) = (ℝ^‘(1...𝑁)) | |
12 | eqid 2651 | . . . . 5 ⊢ (Base‘(ℝ^‘(1...𝑁))) = (Base‘(ℝ^‘(1...𝑁))) | |
13 | 11, 12 | rrxbase 23222 | . . . 4 ⊢ ((1...𝑁) ∈ V → (Base‘(ℝ^‘(1...𝑁))) = {𝑓 ∈ (ℝ ↑𝑚 (1...𝑁)) ∣ 𝑓 finSupp 0}) |
14 | 10, 13 | ax-mp 5 | . . 3 ⊢ (Base‘(ℝ^‘(1...𝑁))) = {𝑓 ∈ (ℝ ↑𝑚 (1...𝑁)) ∣ 𝑓 finSupp 0} |
15 | 9, 14 | eqtr4i 2676 | . 2 ⊢ (ℝ ↑𝑚 (1...𝑁)) = (Base‘(ℝ^‘(1...𝑁))) |
16 | 3, 15 | syl6reqr 2704 | 1 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑𝑚 (1...𝑁)) = (Base‘𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 {crab 2945 Vcvv 3231 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ↑𝑚 cmap 7899 finSupp cfsupp 8316 ℝcr 9973 0cc0 9974 1c1 9975 ℕ0cn0 11330 ...cfz 12364 Basecbs 15904 ℝ^crrx 23217 𝔼hilcehl 23218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-rp 11871 df-fz 12365 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-0g 16149 df-prds 16155 df-pws 16157 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-subg 17638 df-cmn 18241 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-field 18798 df-subrg 18826 df-sra 19220 df-rgmod 19221 df-cnfld 19795 df-refld 19999 df-dsmm 20124 df-frlm 20139 df-tng 22436 df-tch 23015 df-rrx 23219 df-ehl 23220 |
This theorem is referenced by: k0004ss3 38768 |
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