Definition df-rrx 22981

Description: Define the function associating with a set the free real vector space on that set, equipped with the natural inner product. 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 ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))

Detailed syntax breakdown of Definition df-rrx

Step	Hyp	Ref	Expression
1		crrx 22979	. 2 class ℝ^
2		vi	. . 3 setvar 𝑖
3		cvv 3173	. . 3 class V
4		crefld 19769	. . . . 5 class ℝfld
5	2	cv 1474	. . . . 5 class 𝑖
6		cfrlm 19909	. . . . 5 class freeLMod
7	4, 5, 6	co 6549	. . . 4 class (ℝfld freeLMod 𝑖)
8		ctch 22775	. . . 4 class toℂHil
9	7, 8	cfv 5804	. . 3 class (toℂHil‘(ℝfld freeLMod 𝑖))
10	2, 3, 9	cmpt 4643	. 2 class (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
11	1, 10	wceq 1475	1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))

This definition is referenced by:  rrxval  22983