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Theorem rrxval 25335
Description: Value of the generalized Euclidean space. (Contributed by Thierry Arnoux, 16-Jun-2019.)
Hypothesis
Ref Expression
rrxval.r 𝐻 = (ℝ^β€˜πΌ)
Assertion
Ref Expression
rrxval (𝐼 ∈ 𝑉 β†’ 𝐻 = (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝐼)))

Proof of Theorem rrxval
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 rrxval.r . 2 𝐻 = (ℝ^β€˜πΌ)
2 elex 3492 . . 3 (𝐼 ∈ 𝑉 β†’ 𝐼 ∈ V)
3 oveq2 7434 . . . . 5 (𝑖 = 𝐼 β†’ (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
43fveq2d 6906 . . . 4 (𝑖 = 𝐼 β†’ (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝑖)) = (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝐼)))
5 df-rrx 25333 . . . 4 ℝ^ = (𝑖 ∈ V ↦ (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝑖)))
6 fvex 6915 . . . 4 (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝐼)) ∈ V
74, 5, 6fvmpt 7010 . . 3 (𝐼 ∈ V β†’ (ℝ^β€˜πΌ) = (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝐼)))
82, 7syl 17 . 2 (𝐼 ∈ 𝑉 β†’ (ℝ^β€˜πΌ) = (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝐼)))
91, 8eqtrid 2780 1 (𝐼 ∈ 𝑉 β†’ 𝐻 = (toβ„‚PreHilβ€˜(ℝfld freeLMod 𝐼)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473  β€˜cfv 6553  (class class class)co 7426  β„fldcrefld 21543   freeLMod cfrlm 21687  toβ„‚PreHilctcph 25115  β„^crrx 25331
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-rrx 25333
This theorem is referenced by:  rrxbase  25336  rrxprds  25337  rrxnm  25339  rrxcph  25340  rrxds  25341  rrxvsca  25342  rrxplusgvscavalb  25343  rrx0  25345  rrxdim  33345  rrxtopn  45701  opnvonmbllem2  46050
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