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Theorem ehlval 24781
Description: Value of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 16-Jun-2019.)
Hypothesis
Ref Expression
ehlval.e 𝐸 = (𝔼hil𝑁)
Assertion
Ref Expression
ehlval (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))

Proof of Theorem ehlval
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 ehlval.e . 2 𝐸 = (𝔼hil𝑁)
2 oveq2 7366 . . . 4 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
32fveq2d 6847 . . 3 (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁)))
4 df-ehl 24753 . . 3 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
5 fvex 6856 . . 3 (ℝ^‘(1...𝑁)) ∈ V
63, 4, 5fvmpt 6949 . 2 (𝑁 ∈ ℕ0 → (𝔼hil𝑁) = (ℝ^‘(1...𝑁)))
71, 6eqtrid 2789 1 (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6497  (class class class)co 7358  1c1 11053  0cn0 12414  ...cfz 13425  ℝ^crrx 24750  𝔼hilcehl 24751
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-ehl 24753
This theorem is referenced by:  ehlbase  24782  ehl0  24784  ehleudis  24785  ehleudisval  24786  eenglngeehlnm  46832  2sphere  46842  itscnhlinecirc02plem3  46877  inlinecirc02p  46880
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