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Theorem elee 28910
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 7440 . . . . 5 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21oveq2d 7448 . . . 4 (𝑛 = 𝑁 → (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁)))
3 df-ee 28907 . . . 4 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
4 ovex 7465 . . . 4 (ℝ ↑m (1...𝑁)) ∈ V
52, 3, 4fvmpt 7015 . . 3 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑m (1...𝑁)))
65eleq2d 2826 . 2 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁))))
7 reex 11247 . . 3 ℝ ∈ V
8 ovex 7465 . . 3 (1...𝑁) ∈ V
97, 8elmap 8912 . 2 (𝐴 ∈ (ℝ ↑m (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ)
106, 9bitrdi 287 1 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867  cr 11155  1c1 11157  cn 12267  ...cfz 13548  𝔼cee 28904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-ee 28907
This theorem is referenced by:  mptelee  28911  eleei  28913  axlowdimlem5  28962  axlowdimlem7  28964  axlowdimlem10  28967  axlowdimlem14  28971  axlowdim1  28975  elntg2  29001
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