Theorem ehlval 25747

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^‘(0..^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 7435	. . . 4 ⊢ (𝑛 = 𝑁 → (0..^𝑛) = (0..^𝑁))
3	2	fveq2d 6897	. . 3 ⊢ (𝑛 = 𝑁 → (ℝ^‘(0..^𝑛)) = (ℝ^‘(0..^𝑁)))
4		df-ehl 25719	. . 3 ⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(0..^𝑛)))
5		fvex 6906	. . 3 ⊢ (ℝ^‘(0..^𝑁)) ∈ V
6	3, 4, 5	fvmpt 7001	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝔼hil‘𝑁) = (ℝ^‘(0..^𝑁)))
7	1, 6	eqtrid 2810	1 ⊢ (𝑁 ∈ ℕ0 → 𝐸 = (ℝ^‘(0..^𝑁)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  ‘cfv 6546  (class class class)co 7427  0cc0 11210  ℕ₀cn0 12616  ..^cfzo 13798  ℝ^crrx 25716  𝔼_hilcehl 25717

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-ehl 25719

This theorem is used by:  ehlbase  25748  ehl0  25750  ehleudis  25751  ehleudisval  25752  2sphere  50000  itscnhlinecirc02plem3  50035  inlinecirc02p  50038