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Theorem rrxval 25359
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 3480 . . 3 (𝐼𝑉𝐼 ∈ V)
3 oveq2 7427 . . . . 5 (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
43fveq2d 6900 . . . 4 (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
5 df-rrx 25357 . . . 4 ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
6 fvex 6909 . . . 4 (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V
74, 5, 6fvmpt 7004 . . 3 (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
82, 7syl 17 . 2 (𝐼𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
91, 8eqtrid 2777 1 (𝐼𝑉𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  cfv 6549  (class class class)co 7419  fldcrefld 21553   freeLMod cfrlm 21697  toℂPreHilctcph 25139  ℝ^crrx 25355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-rrx 25357
This theorem is referenced by:  rrxbase  25360  rrxprds  25361  rrxnm  25363  rrxcph  25364  rrxds  25365  rrxvsca  25366  rrxplusgvscavalb  25367  rrx0  25369  rrxdim  33443  rrxtopn  45810  opnvonmbllem2  46159
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