Theorem ehlval 23001

Description: Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)

Hypothesis

Ref	Expression
ehlval.e	⊢ 𝐸 = (𝔼hil‘𝑁)

Assertion

Ref	Expression
ehlval	⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁)))

Proof of Theorem ehlval

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ehlval.e	. 2 ⊢ 𝐸 = (𝔼hil‘𝑁)
2		oveq2 6557	. . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
3	2	fveq2d 6107	. . 3 ⊢ (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁)))
4		df-ehl 22982	. . 3 ⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
5		fvex 6113	. . 3 ⊢ (ℝ^‘(1...𝑁)) ∈ V
6	3, 4, 5	fvmpt 6191	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝔼hil‘𝑁) = (ℝ^‘(1...𝑁)))
7	1, 6	syl5eq 2656	1 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁)))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  1c1 9816  ℕ₀cn0 11169  ...cfz 12197  ℝ^crrx 22979  𝔼_hilcehl 22980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-ehl 22982

This theorem is referenced by:  ehlbase  23002