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 e. _V |-> (toCHil ` (RRfld 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 RRfld
5	2	cv 1482	. . . . 5 class i
6		cfrlm 20090	. . . . 5 class freeLMod
7	4, 5, 6	co 6650	. . . 4 class (RRfld freeLMod i )
8		ctch 22967	. . . 4 class toCHil
9	7, 8	cfv 5888	. . 3 class (toCHil ` (RRfld freeLMod i ) )
10	2, 3, 9	cmpt 4729	. 2 class ( i e. _V |-> (toCHil ` (RRfld freeLMod i ) ) )
11	1, 10	wceq 1483	1 wff ℝ^ = ( i e. _V |-> (toCHil ` (RRfld freeLMod i ) ) )

This definition is referenced by:  rrxval  23175