Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rrxval Structured version   Visualization version   GIF version

Theorem rrxval 23969
 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 3489 . . 3 (𝐼𝑉𝐼 ∈ V)
3 oveq2 7138 . . . . 5 (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
43fveq2d 6647 . . . 4 (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
5 df-rrx 23967 . . . 4 ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
6 fvex 6656 . . . 4 (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V
74, 5, 6fvmpt 6741 . . 3 (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
82, 7syl 17 . 2 (𝐼𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
91, 8syl5eq 2868 1 (𝐼𝑉𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  Vcvv 3471  ‘cfv 6328  (class class class)co 7130  ℝfldcrefld 20723   freeLMod cfrlm 20865  toℂPreHilctcph 23750  ℝ^crrx 23965 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-rrx 23967 This theorem is referenced by:  rrxbase  23970  rrxprds  23971  rrxnm  23973  rrxcph  23974  rrxds  23975  rrxvsca  23976  rrxplusgvscavalb  23977  rrx0  23979  rrxdim  31022  rrxtopn  42717  opnvonmbllem2  43063
 Copyright terms: Public domain W3C validator