Definition df-rrx 23173

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 23171	. 2 class ℝ^
2		vi	. . 3 setvar 𝑖
3		cvv 3200	. . 3 class V
4		crefld 19950	. . . . 5 class ℝfld
5	2	cv 1482	. . . . 5 class 𝑖
6		cfrlm 20090	. . . . 5 class freeLMod
7	4, 5, 6	co 6650	. . . 4 class (ℝfld freeLMod 𝑖)
8		ctch 22967	. . . 4 class toℂHil
9	7, 8	cfv 5888	. . 3 class (toℂHil‘(ℝfld freeLMod 𝑖))
10	2, 3, 9	cmpt 4729	. 2 class (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))
11	1, 10	wceq 1483	1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂHil‘(ℝfld freeLMod 𝑖)))

This definition is referenced by:  rrxval  23175