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Theorem rrxval 23917
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 3510 . . 3 (𝐼𝑉𝐼 ∈ V)
3 oveq2 7153 . . . . 5 (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
43fveq2d 6667 . . . 4 (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
5 df-rrx 23915 . . . 4 ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
6 fvex 6676 . . . 4 (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V
74, 5, 6fvmpt 6761 . . 3 (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
82, 7syl 17 . 2 (𝐼𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
91, 8syl5eq 2865 1 (𝐼𝑉𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cfv 6348  (class class class)co 7145  fldcrefld 20676   freeLMod cfrlm 20818  toℂPreHilctcph 23698  ℝ^crrx 23913
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-rrx 23915
This theorem is referenced by:  rrxbase  23918  rrxprds  23919  rrxnm  23921  rrxcph  23922  rrxds  23923  rrxvsca  23924  rrxplusgvscavalb  23925  rrx0  23927  rrxdim  30911  rrxtopn  42446  opnvonmbllem2  42792
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