Theorem rrxval 22983

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	⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂHil‘(ℝfld freeLMod 𝐼)))

Proof of Theorem rrxval

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxval.r	. 2 ⊢ 𝐻 = (ℝ^‘𝐼)
2		elex 3185	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		oveq2 6557	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
4	3	fveq2d 6107	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂHil‘(ℝfld freeLMod 𝑖)) = (toℂHil‘(ℝfld freeLMod 𝐼)))
5		df-rrx 22981	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
6		fvex 6113	. . . 4 ⊢ (toℂHil‘(ℝfld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 6191	. . 3 ⊢ (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂHil‘(ℝfld freeLMod 𝐼)))
8	2, 7	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂHil‘(ℝfld freeLMod 𝐼)))
9	1, 8	syl5eq 2656	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂHil‘(ℝfld freeLMod 𝐼)))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  ℝ_fldcrefld 19769   freeLMod cfrlm 19909  toℂHilctch 22775  ℝ^crrx 22979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-rrx 22981

This theorem is referenced by:  rrxbase  22984  rrxprds  22985  rrxnm  22987  rrxcph  22988  rrxds  22989  rrxtopn  39177  opnvonmbllem2  39523