Definition df-rrx 23990

Description: Define the function associating with a set the free real vector space on that set, equipped with the natural inner product and norm. This is the direct sum of copies of the field of real numbers indexed by that set. We call it here a "generalized real Euclidean space", but note that it need not be complete (for instance if the given set is infinite countable). (Contributed by Thierry Arnoux, 16-Jun-2019.)

Assertion

Ref	Expression
df-rrx	⊢ ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))

Detailed syntax breakdown of Definition df-rrx

Step	Hyp	Ref	Expression
1		crrx 23988	. 2 class ℝ^
2		vi	. . 3 setvar 𝑖
3		cvv 3496	. . 3 class V
4		crefld 20750	. . . . 5 class ℝfld
5	2	cv 1536	. . . . 5 class 𝑖
6		cfrlm 20892	. . . . 5 class freeLMod
7	4, 5, 6	co 7158	. . . 4 class (ℝfld freeLMod 𝑖)
8		ctcph 23773	. . . 4 class toℂPreHil
9	7, 8	cfv 6357	. . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
10	2, 3, 9	cmpt 5148	. 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
11	1, 10	wceq 1537	1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))

This definition is referenced by:  rrxval  23992