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Theorem rrxval 22983
 Description: Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.)
Hypothesis
Ref Expression
rrxval.r 𝐻 = (ℝ^‘𝐼)
Assertion
Ref Expression
rrxval (𝐼𝑉𝐻 = (toℂHil‘(ℝfld freeLMod 𝐼)))

Proof of Theorem rrxval
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 rrxval.r . 2 𝐻 = (ℝ^‘𝐼)
2 elex 3185 . . 3 (𝐼𝑉𝐼 ∈ V)
3 oveq2 6557 . . . . 5 (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
43fveq2d 6107 . . . 4 (𝑖 = 𝐼 → (toℂHil‘(ℝfld freeLMod 𝑖)) = (toℂHil‘(ℝfld freeLMod 𝐼)))
5 df-rrx 22981 . . . 4 ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
6 fvex 6113 . . . 4 (toℂHil‘(ℝfld freeLMod 𝐼)) ∈ V
74, 5, 6fvmpt 6191 . . 3 (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂHil‘(ℝfld freeLMod 𝐼)))
82, 7syl 17 . 2 (𝐼𝑉 → (ℝ^‘𝐼) = (toℂHil‘(ℝfld freeLMod 𝐼)))
91, 8syl5eq 2656 1 (𝐼𝑉𝐻 = (toℂHil‘(ℝfld freeLMod 𝐼)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  ℝfldcrefld 19769   freeLMod cfrlm 19909  toℂHilctch 22775  ℝ^crrx 22979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-rrx 22981 This theorem is referenced by:  rrxbase  22984  rrxprds  22985  rrxnm  22987  rrxcph  22988  rrxds  22989  rrxtopn  39177  opnvonmbllem2  39523
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