Theorem elee 27906

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.

Step	Hyp	Ref	Expression
1		oveq2 7370	. . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2	1	oveq2d 7378	. . . 4 ⊢ (𝑛 = 𝑁 → (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁)))
3		df-ee 27903	. . . 4 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
4		ovex 7395	. . . 4 ⊢ (ℝ ↑m (1...𝑁)) ∈ V
5	2, 3, 4	fvmpt 6953	. . 3 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑m (1...𝑁)))
6	5	eleq2d 2818	. 2 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁))))
7		reex 11151	. . 3 ⊢ ℝ ∈ V
8		ovex 7395	. . 3 ⊢ (1...𝑁) ∈ V
9	7, 8	elmap 8816	. 2 ⊢ (𝐴 ∈ (ℝ ↑m (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ)
10	6, 9	bitrdi 286	1 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362   ↑_m cmap 8772  ℝcr 11059  1c1 11061  ℕcn 12162  ...cfz 13434  𝔼cee 27900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-sbc 3743  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-ee 27903

This theorem is referenced by:  mptelee  27907  eleei  27909  axlowdimlem5  27958  axlowdimlem7  27960  axlowdimlem10  27963  axlowdimlem14  27967  axlowdim1  27971  elntg2  27997