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Theorem elee 26079
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 6854 . . . . 5 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21oveq2d 6862 . . . 4 (𝑛 = 𝑁 → (ℝ ↑𝑚 (1...𝑛)) = (ℝ ↑𝑚 (1...𝑁)))
3 df-ee 26076 . . . 4 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑𝑚 (1...𝑛)))
4 ovex 6878 . . . 4 (ℝ ↑𝑚 (1...𝑁)) ∈ V
52, 3, 4fvmpt 6475 . . 3 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑𝑚 (1...𝑁)))
65eleq2d 2830 . 2 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑𝑚 (1...𝑁))))
7 reex 10284 . . 3 ℝ ∈ V
8 ovex 6878 . . 3 (1...𝑁) ∈ V
97, 8elmap 8093 . 2 (𝐴 ∈ (ℝ ↑𝑚 (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ)
106, 9syl6bb 278 1 (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1652  wcel 2155  wf 6066  cfv 6070  (class class class)co 6846  𝑚 cmap 8064  cr 10192  1c1 10194  cn 11278  ...cfz 12538  𝔼cee 26073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-cnex 10249  ax-resscn 10250
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-map 8066  df-ee 26076
This theorem is referenced by:  mptelee  26080  eleei  26082  axlowdimlem5  26131  axlowdimlem7  26133  axlowdimlem10  26136  axlowdimlem14  26140  axlowdim1  26144
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