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Theorem rrxval 22896
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 3180 . . 3 (𝐼𝑉𝐼 ∈ V)
3 oveq2 6531 . . . . 5 (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
43fveq2d 6088 . . . 4 (𝑖 = 𝐼 → (toℂHil‘(ℝfld freeLMod 𝑖)) = (toℂHil‘(ℝfld freeLMod 𝐼)))
5 df-rrx 22894 . . . 4 ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
6 fvex 6094 . . . 4 (toℂHil‘(ℝfld freeLMod 𝐼)) ∈ V
74, 5, 6fvmpt 6172 . . 3 (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂHil‘(ℝfld freeLMod 𝐼)))
82, 7syl 17 . 2 (𝐼𝑉 → (ℝ^‘𝐼) = (toℂHil‘(ℝfld freeLMod 𝐼)))
91, 8syl5eq 2651 1 (𝐼𝑉𝐻 = (toℂHil‘(ℝfld freeLMod 𝐼)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  Vcvv 3168  cfv 5786  (class class class)co 6523  fldcrefld 19710   freeLMod cfrlm 19847  toℂHilctch 22695  ℝ^crrx 22892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fv 5794  df-ov 6526  df-rrx 22894
This theorem is referenced by:  rrxbase  22897  rrxprds  22898  rrxnm  22900  rrxcph  22901  rrxds  22902  rrxtopn  38977  opnvonmbllem2  39323
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