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Theorem rrxval 25372

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 3452	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
3		oveq2 7364	. . . . 5 ⊢ (𝑖 = 𝐼 → (ℝ_fld freeLMod 𝑖) = (ℝ_fld freeLMod 𝐼))
4	3	fveq2d 6831	. . . 4 ⊢ (𝑖 = 𝐼 → (toℂPreHil‘(ℝ_fld freeLMod 𝑖)) = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))
5		df-rrx 25370	. . . 4 ⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝ_fld freeLMod 𝑖)))
6		fvex 6840	. . . 4 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝐼)) ∈ V
7	4, 5, 6	fvmpt 6935	. . 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 2786	1 ⊢ (𝐼 ∈ 𝑉 → 𝐻 = (toℂPreHil‘(ℝ_fld freeLMod 𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ‘cfv 6485 (class class class)co 7356 ℝ_fldcrefld 21579 freeLMod cfrlm 21721 toℂPreHilctcph 25152 ℝ^crrx 25368

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pr 5362

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-iota 6441 df-fun 6487 df-fv 6493 df-ov 7359 df-rrx 25370

This theorem is referenced by: rrxbase 25373 rrxprds 25374 rrxnm 25376 rrxcph 25377 rrxds 25378 rrxvsca 25379 rrxplusgvscavalb 25380 rrx0 25382 rrxdim 33798 rrxtopn 46727 opnvonmbllem2 47076

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