Theorem ehlval 24931

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 7417	. . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
3	2	fveq2d 6896	. . 3 ⊢ (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁)))
4		df-ehl 24903	. . 3 ⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
5		fvex 6905	. . 3 ⊢ (ℝ^‘(1...𝑁)) ∈ V
6	3, 4, 5	fvmpt 6999	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝔼hil‘𝑁) = (ℝ^‘(1...𝑁)))
7	1, 6	eqtrid 2785	1 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ‘cfv 6544  (class class class)co 7409  1c1 11111  ℕ₀cn0 12472  ...cfz 13484  ℝ^crrx 24900  𝔼_hilcehl 24901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-ehl 24903

This theorem is referenced by:  ehlbase  24932  ehl0  24934  ehleudis  24935  ehleudisval  24936  eenglngeehlnm  47425  2sphere  47435  itscnhlinecirc02plem3  47470  inlinecirc02p  47473