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Theorem elee 25819
Description: Membership in a Euclidean space. We define Euclidean space here using Cartesian coordinates over 𝑁 space. We later abstract away from this using Tarski's geometry axioms, so this exact definition is unimportant. (Contributed by Scott Fenton, 3-Jun-2013.)
Assertion
Ref Expression
elee (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))

Proof of Theorem elee
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . . . 5 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21oveq2d 6706 . . . 4 (𝑛 = 𝑁 → (ℝ ↑𝑚 (1...𝑛)) = (ℝ ↑𝑚 (1...𝑁)))
3 df-ee 25816 . . . 4 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛)))
4 ovex 6718 . . . 4 (ℝ ↑𝑚 (1...𝑁)) ∈ V
52, 3, 4fvmpt 6321 . . 3 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑𝑚 (1...𝑁)))
65eleq2d 2716 . 2 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑𝑚 (1...𝑁))))
7 reex 10065 . . 3 ℝ ∈ V
8 ovex 6718 . . 3 (1...𝑁) ∈ V
97, 8elmap 7928 . 2 (𝐴 ∈ (ℝ ↑𝑚 (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ)
106, 9syl6bb 276 1 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  cr 9973  1c1 9975  cn 11058  ...cfz 12364  𝔼cee 25813
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-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-ee 25816
This theorem is referenced by:  mptelee  25820  eleei  25822  axlowdimlem5  25871  axlowdimlem7  25873  axlowdimlem10  25876  axlowdimlem14  25880  axlowdim1  25884
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