Definition df-frgp 18123

Description: Define the free group on a set 𝐼 of generators, defined as the quotient of the free monoid on 𝐼 × 2_𝑜 (representing the generator elements and their formal inverses) by the free group equivalence relation df-efg 18122. (Contributed by Mario Carneiro, 1-Oct-2015.)

Assertion

Ref	Expression
df-frgp	⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG ‘𝑖)))

Detailed syntax breakdown of Definition df-frgp

Step	Hyp	Ref	Expression
1		cfrgp 18120	. 2 class freeGrp
2		vi	. . 3 setvar 𝑖
3		cvv 3200	. . 3 class V
4	2	cv 1482	. . . . . 6 class 𝑖
5		c2o 7554	. . . . . 6 class 2𝑜
6	4, 5	cxp 5112	. . . . 5 class (𝑖 × 2𝑜)
7		cfrmd 17384	. . . . 5 class freeMnd
8	6, 7	cfv 5888	. . . 4 class (freeMnd‘(𝑖 × 2𝑜))
9		cefg 18119	. . . . 5 class ~FG
10	4, 9	cfv 5888	. . . 4 class ( ~FG ‘𝑖)
11		cqus 16165	. . . 4 class /s
12	8, 10, 11	co 6650	. . 3 class ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG ‘𝑖))
13	2, 3, 12	cmpt 4729	. 2 class (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG ‘𝑖)))
14	1, 13	wceq 1483	1 wff freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜)) /s ( ~FG ‘𝑖)))

This definition is referenced by:  frgpval  18171