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Theorem rrxval 25421
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 = (toℂPreHil‘(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 3501 . . 3 (𝐼𝑉𝐼 ∈ V)
3 oveq2 7439 . . . . 5 (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
43fveq2d 6910 . . . 4 (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
5 df-rrx 25419 . . . 4 ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
6 fvex 6919 . . . 4 (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V
74, 5, 6fvmpt 7016 . . 3 (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
82, 7syl 17 . 2 (𝐼𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
91, 8eqtrid 2789 1 (𝐼𝑉𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cfv 6561  (class class class)co 7431  fldcrefld 21622   freeLMod cfrlm 21766  toℂPreHilctcph 25201  ℝ^crrx 25417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-rrx 25419
This theorem is referenced by:  rrxbase  25422  rrxprds  25423  rrxnm  25425  rrxcph  25426  rrxds  25427  rrxvsca  25428  rrxplusgvscavalb  25429  rrx0  25431  rrxdim  33665  rrxtopn  46299  opnvonmbllem2  46648
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