Definition df-rrx 25599

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 25597	. 2 class ℝ^
2		vi	. . 3 setvar 𝑖
3		cvv 3457	. . 3 class V
4		crefld 21808	. . . . 5 class ℝfld
5	2	cv 1569	. . . . 5 class 𝑖
6		cfrlm 21950	. . . . 5 class freeLMod
7	4, 5, 6	co 7419	. . . 4 class (ℝfld freeLMod 𝑖)
8		ctcph 25381	. . . 4 class toℂPreHil
9	7, 8	cfv 6540	. . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
10	2, 3, 9	cmpt 5194	. 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
11	1, 10	wceq 1570	1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))

This definition is used by:  rrxval  25601