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Theorem ehlval 23944
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 7153 . . . 4 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
32fveq2d 6667 . . 3 (𝑛 = 𝑁 → (ℝ^‘(1...𝑛)) = (ℝ^‘(1...𝑁)))
4 df-ehl 23916 . . 3 𝔼hil = (𝑛 ∈ ℕ0 ↦ (ℝ^‘(1...𝑛)))
5 fvex 6676 . . 3 (ℝ^‘(1...𝑁)) ∈ V
63, 4, 5fvmpt 6761 . 2 (𝑁 ∈ ℕ0 → (𝔼hil𝑁) = (ℝ^‘(1...𝑁)))
71, 6syl5eq 2865 1 (𝑁 ∈ ℕ0𝐸 = (ℝ^‘(1...𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  1c1 10526  0cn0 11885  ...cfz 12880  ℝ^crrx 23913  𝔼hilcehl 23914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-ehl 23916
This theorem is referenced by:  ehlbase  23945  ehl0  23947  ehleudis  23948  ehleudisval  23949  eenglngeehlnm  44654  2sphere  44664  itscnhlinecirc02plem3  44699  inlinecirc02p  44702
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