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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 3465 | . . . 4 ⊢ ((ℝ ↑m (1...𝑁)) = {𝑓 ∈ (ℝ ↑m (1...𝑁)) ∣ 𝑓 finSupp 0} ↔ ∀𝑓 ∈ (ℝ ↑m (1...𝑁))𝑓 finSupp 0) | |
2 | elmapi 8831 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑m (1...𝑁)) → 𝑓:(1...𝑁)⟶ℝ) | |
3 | fzfid 13925 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑m (1...𝑁)) → (1...𝑁) ∈ Fin) | |
4 | 0red 11204 | . . . . 5 ⊢ (𝑓 ∈ (ℝ ↑m (1...𝑁)) → 0 ∈ ℝ) | |
5 | 2, 3, 4 | fdmfifsupp 9361 | . . . 4 ⊢ (𝑓 ∈ (ℝ ↑m (1...𝑁)) → 𝑓 finSupp 0) |
6 | 1, 5 | mprgbir 3069 | . . 3 ⊢ (ℝ ↑m (1...𝑁)) = {𝑓 ∈ (ℝ ↑m (1...𝑁)) ∣ 𝑓 finSupp 0} |
7 | ovex 7429 | . . . 4 ⊢ (1...𝑁) ∈ V | |
8 | eqid 2733 | . . . . 5 ⊢ (ℝ^‘(1...𝑁)) = (ℝ^‘(1...𝑁)) | |
9 | eqid 2733 | . . . . 5 ⊢ (Base‘(ℝ^‘(1...𝑁))) = (Base‘(ℝ^‘(1...𝑁))) | |
10 | 8, 9 | rrxbase 24874 | . . . 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 2764 | . 2 ⊢ (ℝ ↑m (1...𝑁)) = (Base‘(ℝ^‘(1...𝑁))) |
13 | ehlval.e | . . . 4 ⊢ 𝐸 = (𝔼hil‘𝑁) | |
14 | 13 | ehlval 24900 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁))) |
15 | 14 | fveq2d 6885 | . 2 ⊢ (𝑁 ∈ ℕ0 → (Base‘𝐸) = (Base‘(ℝ^‘(1...𝑁)))) |
16 | 12, 15 | eqtr4id 2792 | 1 ⊢ (𝑁 ∈ ℕ0 → (ℝ ↑m (1...𝑁)) = (Base‘𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 class class class wbr 5144 ‘cfv 6535 (class class class)co 7396 ↑m cmap 8808 finSupp cfsupp 9349 ℝcr 11096 0cc0 11097 1c1 11098 ℕ0cn0 12459 ...cfz 13471 Basecbs 17131 ℝ^crrx 24869 𝔼hilcehl 24870 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 ax-addf 11176 ax-mulf 11177 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-supp 8134 df-tpos 8198 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9350 df-sup 9424 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-rp 12962 df-fz 13472 df-seq 13954 df-exp 14015 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-starv 17199 df-sca 17200 df-vsca 17201 df-ip 17202 df-tset 17203 df-ple 17204 df-ds 17206 df-unif 17207 df-hom 17208 df-cco 17209 df-0g 17374 df-prds 17380 df-pws 17382 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-grp 18809 df-minusg 18810 df-subg 18988 df-cmn 19634 df-mgp 19971 df-ur 19988 df-ring 20040 df-cring 20041 df-oppr 20128 df-dvdsr 20149 df-unit 20150 df-invr 20180 df-dvr 20193 df-drng 20295 df-field 20296 df-subrg 20338 df-sra 20762 df-rgmod 20763 df-cnfld 20919 df-refld 21131 df-dsmm 21260 df-frlm 21275 df-tng 24062 df-tcph 24655 df-rrx 24871 df-ehl 24872 |
This theorem is referenced by: ehl0base 24902 k0004ss3 42775 |
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