Definition df-rrx 25419

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 25417	. 2 class ℝ^
2		vi	. . 3 setvar 𝑖
3		cvv 3480	. . 3 class V
4		crefld 21622	. . . . 5 class ℝfld
5	2	cv 1539	. . . . 5 class 𝑖
6		cfrlm 21766	. . . . 5 class freeLMod
7	4, 5, 6	co 7431	. . . 4 class (ℝfld freeLMod 𝑖)
8		ctcph 25201	. . . 4 class toℂPreHil
9	7, 8	cfv 6561	. . 3 class (toℂPreHil‘(ℝfld freeLMod 𝑖))
10	2, 3, 9	cmpt 5225	. 2 class (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))
11	1, 10	wceq 1540	1 wff ℝ^ = (𝑖 ∈ V ↦ (toℂPreHil‘(ℝfld freeLMod 𝑖)))

This definition is referenced by:  rrxval  25421