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Theorem rrxval 25337

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 7411	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝ_fld freeLMod 𝑖) = (ℝ_fld freeLMod 𝐼))
4	3	fveq2d 6879	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝ_fld freeLMod 𝑖)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5		df-rrx 25335	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝ_fld freeLMod 𝑖)))
6		fvex 6888	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 6985	. . 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 2782	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3459 ‘cfv 6530 (class class class)co 7403 ℝ_fldcrefld 21562 freeLMod cfrlm 21704 toℂPreHilctcph 25117 ℝ^crrx 25333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-iota 6483 df-fun 6532 df-fv 6538 df-ov 7406 df-rrx 25335

This theorem is referenced by: rrxbase 25338 rrxprds 25339 rrxnm 25341 rrxcph 25342 rrxds 25343 rrxvsca 25344 rrxplusgvscavalb 25345 rrx0 25347 rrxdim 33600 rrxtopn 46261 opnvonmbllem2 46610

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