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Theorem elee 28142
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 7414 . . . . 5 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21oveq2d 7422 . . . 4 (𝑛 = 𝑁 → (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁)))
3 df-ee 28139 . . . 4 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
4 ovex 7439 . . . 4 (ℝ ↑m (1...𝑁)) ∈ V
52, 3, 4fvmpt 6996 . . 3 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑m (1...𝑁)))
65eleq2d 2820 . 2 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁))))
7 reex 11198 . . 3 ℝ ∈ V
8 ovex 7439 . . 3 (1...𝑁) ∈ V
97, 8elmap 8862 . 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 205   = wceq 1542  wcel 2107  wf 6537  cfv 6541  (class class class)co 7406  m cmap 8817  cr 11106  1c1 11108  cn 12209  ...cfz 13481  𝔼cee 28136
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-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-map 8819  df-ee 28139
This theorem is referenced by:  mptelee  28143  eleei  28145  axlowdimlem5  28194  axlowdimlem7  28196  axlowdimlem10  28199  axlowdimlem14  28203  axlowdim1  28207  elntg2  28233
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