Theorem elee 28978

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 7376	. . . . 5 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2	1	oveq2d 7384	. . . 4 ⊢ (𝑛 = 𝑁 → (ℝ ↑m (1...𝑛)) = (ℝ ↑m (1...𝑁)))
3		df-ee 28975	. . . 4 ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ ↑m (1...𝑛)))
4		ovex 7401	. . . 4 ⊢ (ℝ ↑m (1...𝑁)) ∈ V
5	2, 3, 4	fvmpt 6949	. . 3 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (ℝ ↑m (1...𝑁)))
6	5	eleq2d 2823	. 2 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴 ∈ (ℝ ↑m (1...𝑁))))
7		reex 11129	. . 3 ⊢ ℝ ∈ V
8		ovex 7401	. . 3 ⊢ (1...𝑁) ∈ V
9	7, 8	elmap 8821	. 2 ⊢ (𝐴 ∈ (ℝ ↑m (1...𝑁)) ↔ 𝐴:(1...𝑁)⟶ℝ)
10	6, 9	bitrdi 287	1 ⊢ (𝑁 ∈ ℕ → (𝐴 ∈ (𝔼‘𝑁) ↔ 𝐴:(1...𝑁)⟶ℝ))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775  ℝcr 11037  1c1 11039  ℕcn 12157  ...cfz 13435  𝔼cee 28972

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-ee 28975

This theorem is referenced by:  mptelee  28979  mpteleeOLD  28980  eleei  28982  axlowdimlem5  29031  axlowdimlem7  29033  axlowdimlem10  29036  axlowdimlem14  29040  axlowdim1  29044  elntg2  29070