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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 | rabid2 3307 | . . . 4 ⊢ ((ℝ ↑m (1...𝑁)) = {𝑓 ∈ (ℝ ↑m (1...𝑁)) ∣ 𝑓 finSupp 0} ↔ ∀𝑓 ∈ (ℝ ↑m (1...𝑁))𝑓 finSupp 0) | |
2 | elmapi 8595 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑m (1...𝑁)) → 𝑓:(1...𝑁)⟶ℝ) | |
3 | fzfid 13621 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑m (1...𝑁)) → (1...𝑁) ∈ Fin) | |
4 | 0red 10909 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑m (1...𝑁)) → 0 ∈ ℝ) | |
5 | 2, 3, 4 | fdmfifsupp 9068 | . . . 4 ⊢ (𝑓 ∈ (ℝ ↑m (1...𝑁)) → 𝑓 finSupp 0) |
6 | 1, 5 | mprgbir 3078 | . . 3 ⊢ (ℝ ↑m (1...𝑁)) = {𝑓 ∈ (ℝ ↑m (1...𝑁)) ∣ 𝑓 finSupp 0} |
7 | ovex 7288 | . . . 4 ⊢ (1...𝑁) ∈ V | |
8 | eqid 2738 | . . . . 5 ⊢ (ℝ^‘(1...𝑁)) = (ℝ^‘(1...𝑁)) | |
9 | eqid 2738 | . . . . 5 ⊢ (Base‘(ℝ^‘(1...𝑁))) = (Base‘(ℝ^‘(1...𝑁))) | |
10 | 8, 9 | rrxbase 24457 | . . . 4 ⊢ ((1...𝑁) ∈ V → (Base‘(ℝ^‘(1...𝑁))) = {𝑓 ∈ (ℝ ↑m (1...𝑁)) ∣ 𝑓 finSupp 0}) |
11 | 7, 10 | ax-mp 5 | . . 3 ⊢ (Base‘(ℝ^‘(1...𝑁))) = {𝑓 ∈ (ℝ ↑m (1...𝑁)) ∣ 𝑓 finSupp 0} |
12 | 6, 11 | eqtr4i 2769 | . 2 ⊢ (ℝ ↑m (1...𝑁)) = (Base‘(ℝ^‘(1...𝑁))) |
13 | ehlval.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘𝑁) | |
14 | 13 | ehlval 24483 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) |
15 | 14 | fveq2d 6760 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Base‘𝐸) = (Base‘(ℝ^‘(1...𝑁)))) |
16 | 12, 15 | eqtr4id 2798 | 1 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑m (1...𝑁)) = (Base‘𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 {crab 3067 Vcvv 3422 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ↑m cmap 8573 finSupp cfsupp 9058 ℝcr 10801 0cc0 10802 1c1 10803 ℕ0cn0 12163 ...cfz 13168 Basecbs 16840 ℝ^crrx 24452 𝔼hilcehl 24453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-0g 17069 df-prds 17075 df-pws 17077 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496 df-subg 18667 df-cmn 19303 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-field 19909 df-subrg 19937 df-sra 20349 df-rgmod 20350 df-cnfld 20511 df-refld 20722 df-dsmm 20849 df-frlm 20864 df-tng 23646 df-tcph 24238 df-rrx 24454 df-ehl 24455 |
This theorem is referenced by: ehl0base 24485 k0004ss3 41652 |
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