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Theorem ehlbase 25156

Description: The base of the Euclidean space is the set of n-tuples of real numbers. (Contributed by Thierry Arnoux, 16-Jun-2019.)

**Hypothesis**
Ref	Expression
ehlval.e	⊢ 𝐸 = (𝔼_hil‘𝑁)

**Assertion**
Ref	Expression
ehlbase	⊢ (𝑁 ∈ ℕ₀ → (ℝ ↑_m (1...𝑁)) = (Base‘𝐸))

Proof of Theorem ehlbase Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabid2 3464	. . . 4 ⊢ ((ℝ ↑_m (1...𝑁)) = {𝑓 ∈ (ℝ ↑_m (1...𝑁)) ∣ 𝑓 finSupp 0} ↔ ∀𝑓 ∈ (ℝ ↑_m (1...𝑁))𝑓 finSupp 0)
2		elmapi 8845	. . . . 5 ⊢ (𝑓 ∈ (ℝ ↑_m (1...𝑁)) → 𝑓:(1...𝑁)⟶ℝ)
3		fzfid 13942	. . . . 5 ⊢ (𝑓 ∈ (ℝ ↑_m (1...𝑁)) → (1...𝑁) ∈ Fin)
4		0red 11221	. . . . 5 ⊢ (𝑓 ∈ (ℝ ↑_m (1...𝑁)) → 0 ∈ ℝ)
5	2, 3, 4	fdmfifsupp 9375	. . . 4 ⊢ (𝑓 ∈ (ℝ ↑_m (1...𝑁)) → 𝑓 finSupp 0)
6	1, 5	mprgbir 3068	. . 3 ⊢ (ℝ ↑_m (1...𝑁)) = {𝑓 ∈ (ℝ ↑_m (1...𝑁)) ∣ 𝑓 finSupp 0}
7		ovex 7444	. . . 4 ⊢ (1...𝑁) ∈ V
8		eqid 2732	. . . . 5 ⊢ (ℝ^‘(1...𝑁)) = (ℝ^‘(1...𝑁))
9		eqid 2732	. . . . 5 ⊢ (Base‘(ℝ^‘(1...𝑁))) = (Base‘(ℝ^‘(1...𝑁)))
10	8, 9	rrxbase 25129	. . . 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 2763	. 2 ⊢ (ℝ ↑_m (1...𝑁)) = (Base‘(ℝ^‘(1...𝑁)))
13		ehlval.e	. . . 4 ⊢ 𝐸 = (𝔼_hil‘𝑁)
14	13	ehlval 25155	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝐸 = (ℝ^‘(1...𝑁)))
15	14	fveq2d 6895	. 2 ⊢ (𝑁 ∈ ℕ₀ → (Base‘𝐸) = (Base‘(ℝ^‘(1...𝑁))))
16	12, 15	eqtr4id 2791	1 ⊢ (𝑁 ∈ ℕ₀ → (ℝ ↑_m (1...𝑁)) = (Base‘𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 ↑_m cmap 8822 finSupp cfsupp 9363 ℝcr 11111 0cc0 11112 1c1 11113 ℕ₀cn0 12476 ...cfz 13488 Basecbs 17148 ℝ^crrx 25124 𝔼_hilcehl 25125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12979 df-fz 13489 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-0g 17391 df-prds 17397 df-pws 17399 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-subg 19039 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-subrng 20434 df-subrg 20459 df-drng 20502 df-field 20503 df-sra 20930 df-rgmod 20931 df-cnfld 21145 df-refld 21377 df-dsmm 21506 df-frlm 21521 df-tng 24313 df-tcph 24910 df-rrx 25126 df-ehl 25127

This theorem is referenced by: ehl0base 25157 k0004ss3 43206

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