Theorem ehlval 25542

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 7419	. . . 4 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
3	2	fveq2d 6886	. . 3 ⊢ (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁)))
4		df-ehl 25514	. . 3 ⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
5		fvex 6895	. . 3 ⊢ (ℝ^‘(1...𝑁)) ∈ V
6	3, 4, 5	fvmpt 6990	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝔼hil‘𝑁) = (ℝ^‘(1...𝑁)))
7	1, 6	eqtrid 2816	1 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(1...𝑁)))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  (class class class)co 7411  1c1 11101  ℕ₀cn0 12504  ...cfz 13535  ℝ^crrx 25511  𝔼_hilcehl 25512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-ehl 25514

This theorem is referenced by:  ehlbase  25543  ehl0  25545  ehleudis  25546  ehleudisval  25547  eenglngeehlnm  49404  2sphere  49414  itscnhlinecirc02plem3  49449  inlinecirc02p  49452