MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elee Structured version   Visualization version   GIF version

Theorem elee 28924
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 7439 . . . . 5 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21oveq2d 7447 . . . 4 (𝑛 = 𝑁 → (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁)))
3 df-ee 28921 . . . 4 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
4 ovex 7464 . . . 4 (ℝ ↑m (1...𝑁)) ∈ V
52, 3, 4fvmpt 7016 . . 3 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑m (1...𝑁)))
65eleq2d 2825 . 2 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁))))
7 reex 11244 . . 3 ℝ ∈ V
8 ovex 7464 . . 3 (1...𝑁) ∈ V
97, 8elmap 8910 . 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 1537  wcel 2106  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  cr 11152  1c1 11154  cn 12264  ...cfz 13544  𝔼cee 28918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-ee 28921
This theorem is referenced by:  mptelee  28925  eleei  28927  axlowdimlem5  28976  axlowdimlem7  28978  axlowdimlem10  28981  axlowdimlem14  28985  axlowdim1  28989  elntg2  29015
  Copyright terms: Public domain W3C validator