Definition df-frgp 19577

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 19576. (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 19574	. 2 class freeGrp
2		vi	. . 3 setvar 𝑖
3		cvv 3474	. . 3 class V
4	2	cv 1540	. . . . . 6 class 𝑖
5		c2o 8459	. . . . . 6 class 2o
6	4, 5	cxp 5674	. . . . 5 class (𝑖 × 2o)
7		cfrmd 18727	. . . . 5 class freeMnd
8	6, 7	cfv 6543	. . . 4 class (freeMnd‘(𝑖 × 2o))
9		cefg 19573	. . . . 5 class ~FG
10	4, 9	cfv 6543	. . . 4 class ( ~FG ‘𝑖)
11		cqus 17450	. . . 4 class /s
12	8, 10, 11	co 7408	. . 3 class ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖))
13	2, 3, 12	cmpt 5231	. 2 class (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))
14	1, 13	wceq 1541	1 wff freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2o)) /s ( ~FG ‘𝑖)))

This definition is referenced by:  frgpval  19625