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Theorem ehlval 23620
 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 6930 . . . 4 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
32fveq2d 6450 . . 3 (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁)))
4 df-ehl 23592 . . 3 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
5 fvex 6459 . . 3 (ℝ^‘(1...𝑁)) ∈ V
63, 4, 5fvmpt 6542 . 2 (𝑁 ∈ ℕ0 → (𝔼hil𝑁) = (ℝ^‘(1...𝑁)))
71, 6syl5eq 2826 1 (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  ‘cfv 6135  (class class class)co 6922  1c1 10273  ℕ0cn0 11642  ...cfz 12643  ℝ^crrx 23589  𝔼hilcehl 23590 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-ehl 23592 This theorem is referenced by:  ehlbase  23621  ehl0  23623  ehleudis  23624  ehleudisval  23625  eenglngeehlnm  43475  2sphere  43485  itscnhlinecirc02plem3  43520  inlinecirc02p  43523
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