Definition df-rrx 25312

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 25310	. 2 class ℝ^
2		vi	. . 3 setvar 𝑖
3		cvv 3436	. . 3 class V
4		crefld 21541	. . . . 5 class ℝfld
5	2	cv 1540	. . . . 5 class 𝑖
6		cfrlm 21683	. . . . 5 class freeLMod
7	4, 5, 6	co 7346	. . . 4 class (ℝfld freeLMod 𝑖)
8		ctcph 25094	. . . 4 class toℂPreHil
9	7, 8	cfv 6481	. . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
10	2, 3, 9	cmpt 5170	. 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
11	1, 10	wceq 1541	1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))

This definition is referenced by:  rrxval  25314