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Theorem rrxval 25341

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‘(ℝ_fld 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 3459	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		oveq2 7364	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝ_fld freeLMod 𝑖) = (ℝ_fld freeLMod 𝐼))
4	3	fveq2d 6836	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝ_fld freeLMod 𝑖)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5		df-rrx 25339	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝ_fld freeLMod 𝑖)))
6		fvex 6845	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 6939	. . 3 ⊢ (𝐼 ∈ V → (ℝ^‘𝐼) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
8	2, 7	syl 17	. 2 ⊢ (𝐼 ∈ 𝑉 → (ℝ^‘𝐼) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
9	1, 8	eqtrid 2781	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 Vcvv 3438 ‘cfv 6490 (class class class)co 7356 ℝ_fldcrefld 21557 freeLMod cfrlm 21699 toℂPreHilctcph 25121 ℝ^crrx 25337

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-dif 3902 df-un 3904 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-iota 6446 df-fun 6492 df-fv 6498 df-ov 7359 df-rrx 25339

This theorem is referenced by: rrxbase 25342 rrxprds 25343 rrxnm 25345 rrxcph 25346 rrxds 25347 rrxvsca 25348 rrxplusgvscavalb 25349 rrx0 25351 rrxdim 33720 rrxtopn 46470 opnvonmbllem2 46819

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