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Theorem rrxval 25359

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 3480	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		oveq2 7427	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝ_fld freeLMod 𝑖) = (ℝ_fld freeLMod 𝐼))
4	3	fveq2d 6900	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝ_fld freeLMod 𝑖)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5		df-rrx 25357	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝ_fld freeLMod 𝑖)))
6		fvex 6909	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 7004	. . 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 2777	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ‘cfv 6549 (class class class)co 7419 ℝ_fldcrefld 21553 freeLMod cfrlm 21697 toℂPreHilctcph 25139 ℝ^crrx 25355

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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-rrx 25357

This theorem is referenced by: rrxbase 25360 rrxprds 25361 rrxnm 25363 rrxcph 25364 rrxds 25365 rrxvsca 25366 rrxplusgvscavalb 25367 rrx0 25369 rrxdim 33443 rrxtopn 45810 opnvonmbllem2 46159

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