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Theorem ehlval 25462
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 7439 . . . 4 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
32fveq2d 6911 . . 3 (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁)))
4 df-ehl 25434 . . 3 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
5 fvex 6920 . . 3 (ℝ^‘(1...𝑁)) ∈ V
63, 4, 5fvmpt 7016 . 2 (𝑁 ∈ ℕ0 → (𝔼hil𝑁) = (ℝ^‘(1...𝑁)))
71, 6eqtrid 2787 1 (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  1c1 11154  0cn0 12524  ...cfz 13544  ℝ^crrx 25431  𝔼hilcehl 25432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-ehl 25434
This theorem is referenced by:  ehlbase  25463  ehl0  25465  ehleudis  25466  ehleudisval  25467  eenglngeehlnm  48589  2sphere  48599  itscnhlinecirc02plem3  48634  inlinecirc02p  48637
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