Theorem ehlval 25476

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 7404	. . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
3	2	fveq2d 6871	. . 3 ⊢ (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁)))
4		df-ehl 25448	. . 3 ⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
5		fvex 6880	. . 3 ⊢ (ℝ^‘(1...𝑁)) ∈ V
6	3, 4, 5	fvmpt 6975	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝔼hil‘𝑁) = (ℝ^‘(1...𝑁)))
7	1, 6	eqtrid 2809	1 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁)))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  ‘cfv 6521  (class class class)co 7396  1c1 11074  ℕ₀cn0 12481  ...cfz 13512  ℝ^crrx 25445  𝔼_hilcehl 25446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-ehl 25448

This theorem is referenced by:  ehlbase  25477  ehl0  25479  ehleudis  25480  ehleudisval  25481  eenglngeehlnm  49361  2sphere  49371  itscnhlinecirc02plem3  49406  inlinecirc02p  49409