Theorem rrxval 25266

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 = (toℂPreHil‘(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 3487	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		oveq2 7412	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝfld freeLMod 𝑖) = (ℝfld freeLMod 𝐼))
4	3	fveq2d 6888	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝfld freeLMod 𝑖)) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
5		df-rrx 25264	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
6		fvex 6897	. . . 4 ⊢ (toℂPreHil‘(ℝfld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 6991	. . 3 ⊢ (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
8	2, 7	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝfld freeLMod 𝐼)))
9	1, 8	eqtrid 2778	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝfld freeLMod 𝐼)))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ‘cfv 6536  (class class class)co 7404  ℝ_fldcrefld 21493   freeLMod cfrlm 21637  toℂPreHilctcph 25046  ℝ^crrx 25262

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-rrx 25264

This theorem is referenced by:  rrxbase  25267  rrxprds  25268  rrxnm  25270  rrxcph  25271  rrxds  25272  rrxvsca  25273  rrxplusgvscavalb  25274  rrx0  25276  rrxdim  33217  rrxtopn  45553  opnvonmbllem2  45902